Group Theory
Hartmut Laue
Mathematisches Seminar der Universität Kiel 2013
Preface
These lecture notes present the contents of my course on Group Theory within the
masters programme in Mathematics at the University of Kiel. The aim is to introduce
into concepts and techniques of modern group theory which are the prerequisites for
tackling current research problems. In an area which has been studied with extreme
intensity for many decades, the decision of what to include or not under the time limits
of a summer semester was certainly not trivial, and apart from the aspect of importance
also that of personal taste had to play a role.
Experts will soon discover that among the results proved in this course there are certain
theorems which frequently are viewed as too difficult to reach, like Tate’s (4.10) or
Roquette’s (5.13). The proofs given here need only a few lines thanks to an approach
which seems to have been underestimated although certain rudiments of it have made
it into newer textbooks. Instead of making heavy use of cohomological or topological
considerations or character theory, we introduce a completely elementary but rather
general concept of normalized group action (1.5.4) which serves as a base for not only the
above-mentioned highlights but also for other important theorems (3.6, 3.9 (Gaschütz),
3.13 (Schur-Zassenhaus)) and for the transfer.
Thus we hope to escape the cartesian reservation towards authors in general1 , although
other parts of the theory clearly follow well-known patterns when a major modification
would not result in a gain of clarity or applicability. Nevertheless, a closer look shows
that details frequently differ from classical expositions. The reader is urged to consult
these, to compare, to develop his own understanding and view. A major difference to
traditional presentations will be observed with respect to the Krull-Schmidt theorem
which is proved in a general version (2.5) containing as special cases the classical one
(characterized by the hypothesis of chain conditions) and the form for modules due to
Azumaya. Throughout these notes we avoid to suppose finiteness where not necessary.
On the other hand, the spirit of the text is clearly determined by central developments
in the theory of finite groups.
The course does not start at the level of the definition of a group. Knowledge of certain
basics, usually provided by a standard introductory course in Algebra, is assumed. But
1
“[. . . ]Authors are ordinarily so disposed that whenever their heedless credulity has led them to a
decision on some controverted opinion, they always try to bring us over to the same side, with the
subtlest arguments; if on the other hand they have been fortunate enough to discover something
certain and evident, they never set it forth without wrapping it up in all sorts of complications. I
suppose they are afraid that a simple account may lessen the importance they gain by the discovery;
or perhaps they begrudge us the plain truth.” Descartes, Reg. 3 (transl. E. Anscombe, P. T. Geach,
Descartes Philosophical Writings, London 1954.)
1
apart from simplest standard topics, the course is self-contained. A detailed list of what
will be assumed to be known may be found on p. 4. It is not necessary as a prerequisite
to be informed about deeper properties of soluble groups and solubility criteria. These
topics are essentially independent of and rather a supplement to these lecture notes, but
of course highly recommended for a broader background. We think that a reader who
feels attracted, hopefully even fascinated by groups will – and should – in any case study
group rings, representations, characters, subjects which are not covered by the present
text. The basics of these are treated, for example, in my course on solubility of equations
and groups (currently running under the label “Algebra II” as part of the bachelor degree
programme). Of course, there are more than enough specialized textbooks on these
topics for readers who want to study them in greater detail.
Kiel, July 2013
Hartmut Laue
2
Contents
1 Permutations and group actions
5
2 Groups with operators
23
3 Complements
34
4 Transfer
48
5 Nilpotency
54
A Appendix and Outlook
68
Prerequisites
Readers should be familiar with the notions
group, subgroup, normal subgroup, quotient (factor group), index, cyclic group, abelian
group, soluble group, simple group, commutator, commutator subgroup of a group,
symmetric group, sign homomorphism.
It is assumed that the homomorphism theorem for groups is at the reader’s disposal,
also the isomorphism theorems as consequences; in particular, it should be known how
the subgroups of a factor group of G are obtained from subgroups of G. In general,
the product set of two subgroups need not be a subgroup. But this is the case if one
of the two subgroups is normal. The product of two normal subgroups is not only a
subgroup but even normal. The order of a subgroup of a finite group G is a divisor of
|G| (Lagrange’s theorem), and G is cyclic if and only if for each divisor d of |G| there
exists exactly one subgroup of G of order d.
A further most useful general proposition is the so-called extension principle a proof
of which will not be given here: Let ϕ be an injective mapping of a set B into a set
M. Then there exists a set B̂ containing B and a bijection ϕ̂ of B̂ onto M such that
ϕ̂|B = ϕ. If ◦ is an operation on B, · an operation on M such that ϕ is a homomorphism
with respect to these operations, then ◦ extends to an operation ˆ◦ of B̂ such that ϕ̂ is
an isomorphism of (B̂, ˆ◦) onto (M, ·).
We write N for the set of all positive integers and set N0 := N ∪ {0}. For all n ∈ N0 we
put
n := {k|k ∈ N, k ≤ n}.
Ṡ
A dot above the symbol of a union ( ) is used if the union is disjoint. If G is a group,
we write G′ for its commutator subgroup. Furthermore,  stands for the trivial subgroup
{1G }.
4
1 Permutations and group actions
If X is a set and V is a set of operations on X, a bijection of X onto X which is a
homomorphism with respect to each operation in V is called an automorphism of the
structure (X, V). We write Aut (X, V) for the set of all automorphisms of (X, V). Recall
that Aut (X, V) is a group with respect to the natural composition of mappings, called
the automorphism group of (X, V). Usually it is clear from the context which operations
on the set X are considered, i. e., there is no doubt about the set V: If we study groups,
V will just contain a single element, the group operation; if we study fields, V will consist
of the addition and the multiplication of the field in question, etc. If there is no doubt
about the meaning of V we speak simply of automorphisms of X and write Aut X for
its automorphism group.
Galois theory is a convincing and ground-laying example of the general idea to study
structures by means of an analysis of their automorphism group. The trivial remark
that the automorphisms of any structure always form a group explains an important
universal aspect of the notion of group: Studying an arbitrary structure, describing the
structural roles of its elements, leads and amounts to studying its automorphism group.
This principle may even be applied to groups themselves as a special class of algebraic
structures. Every element g of a group G induces a so-called inner automorphism ḡ of G
by means of conjugation:
ḡ :
G → G,
x 7→ xg := g −1 xg,
is an automorphism of G, and the mapping
κ : G → Aut G,
g 7→ ḡ,
is a homomorphism whose kernel Z(G) is called the centre of G. We have the following
4-fold description of its elements:
g ∈ Z(G) ⇔ ∀x ∈ G xg = x ⇔ ∀x ∈ G xg = gx ⇔ ∀x ∈ G g = g x
which allows the following three-fold interpretation of Z(G):
1.0.1. For every group G, Z(G) is the set of all elements of G
– the conjugation by which induces the identity on G,
– which commute with every element of G,
– which are left fixed under the conjugation by an arbitrary element of G.
5
In particular, Z(G) is an abelian normal subgroup of G. More generally, for every
x ∈ G, the set Z(G) ∪ {x} consists of mutually commuting elements, hence generates an
abelian subgroup. If we assume that G/Z(G) is cyclic, we may choose x as an element
of a generator of G/Z(G) and obtain:
1.0.2. If G/Z(G) is cyclic, G is abelian (i. e., G/Z(G) = ).
For all g ∈ G, α ∈ Aut G we have2 g α = g α . Hence the image of G under κ is a normal
subgroup of Aut G. It is denoted by In G and called the inner automorphism group of G.
Thus, making use of the homomorphism theorem, we obtain
G/Z(G) ∼
= In G E Aut G for every group G.
A subset X of a group G is called normal if X g = X for all g ∈ G.3 If X is a normal
subset of G, then so is G r X; in particular, G r {1G } is normal. For every x ∈ G, the
set
xG := {xg |g ∈ G}
is the smallest normal subset of G containing x and is called the conjugacy class of x in
G. From the third description of Z(G) in 1.0.1 we obtain
1.0.3. For every element x of a group G, x ∈ Z(G) ⇔ xG = {x} ⇔ |xG | = 1.
Let ∼ be the relation on a group G defined by
G
x ∼ y ↔ ∃g ∈ G xg = y,
G
for all x, y ∈ G.
1.0.4. ∼ is an equivalence relation on G. The equivalence classes are exactly the conG
jugacy classes of G. In particular, the conjugacy classes of G form a partition of G. An important special case of an automorphism group Aut (X, V) arises if V = ∅ : We
have Aut (X, ∅) = SX , the symmetric group on X, consisting of all bijections of X onto
X. We shall now make several observations about our general notions in this special
type of group. We confine ourselves to the case of a finite set X.
Let π ∈ SX . The relation ∼ on X defined by
π
m ∼ m′ ↔ ∃k ∈ Z mπ k = m′ ,
π
for all m, m′ ∈ X,
is an equivalence relation. In particular, its equivalence classes form a partition of X.
1.1 Definition. Let X be finite set and π ∈ SX . An equivalence class of ∼ is called a
π
π-orbit in X and nontrivial if it contains more than one element. If there is at most one
nontrivial π-orbit, π is called a cycle.
2
For the image of an element of a multiplicatively written group under a homomorphism, the exponential notation is convenient. This explains the meaning of g α while g α merely is a particular case
of conjugation, within the group Aut G.
3
equivalently, if X g ⊆ X for all g ∈ G, i. e., if X is invariant under all conjugations.
6
Obviously, idX is the only permutation for which each orbit is trivial. Now let ζ ∈ SX
be a cycle 6= idX and m ∈ X such that mζ 6= m. If k is the order4 of ζ, we set
m1 := m,
(∗)
∀j ∈ k r {1} mj := mj−1 ζ.
Then mj = mζ j−1 for all j ∈ k ; {m1 , . . . , mk } is the nontrivial ζ-orbit and contains
exactly k elements. By (∗), the k-tuple (m1 , . . . , mk ) determines uniquely the cycle ζ.
It is customary to write the cycle ζ “abusing” the notation of an 1 × k matrix:
ζ = m1 . . . mk
The choice of m = m1 was arbitrary within the nontrivial ζ-orbit so that any of the k
elements mj could take its place. We have
m1 m2 . . . mk = m2 . . . mk m1 = · · · = mk m1 . . . mk−1
which, of course, makes sense only as series of equalities of cycles, not of 1 × k matrices.
This notation is the reason for calling k the length of the cycle ζ. A cycle of length
k is called a k-cycle. Cycles ζ, ζ ′ in SX are called disjoint if their nontrivial orbits (if
existent) are disjoint. If this is the case, then each m ∈ X is fixed by at least one of the
two cycles. In particular, mζζ ′ = mζ ′ζ for all m ∈ X, i. e., ζ, ζ ′ commute. We conclude:
1.1.1. Any composition of mutually disjoint cycles is independent of the choice of the
order of the factors.
Q
Thus, for any set C of mutually disjoint cycles 6= idX in SX , the product π = C is
well-defined. If m ∈ X and mπ 6= m, there exists a unique ζ ∈ C such that mζ 6= m.
More precisely, we then have mπ j = mζ j Q
for all j ∈ Z. Hence each ζ ∈ C is uniquely
determined by π. The mapping p : C 7→ C thus is an injection of the set CX of all
sets of mutually disjoint cycles 6= idX into SX . In fact, p is a bijection: For an arbitrary
π ∈ SX we choose a set of representatives R for the set of all non-trivial π-orbits in X
and define, for each r ∈ R, ζr to be the cycle with the property
rζ j = rπ j
Then
Q
r∈R ζr
for all j ∈ Z.
= π. We have proved:
1.1.2. ForQeach π ∈ SX there exists a unique set C of mutually disjoint cycles in SX ridX
such that C = π.
The set C Q
associated with π is called the cycle decomposition of π. For all n ∈ N we
n
have π = ζ∈C ζ n , by 1.1.1. This shows, in particular,
1.1.3. If π ∈ SX has the cycle decomposition C, then o(π) = lcm{o(ζ)|ζ ∈ C}.
Q
Furthermore, π σ = ζ∈C ζ σ for all σ ∈ SX . The first part of the following remark shows
that each ζ σ is again a cycle the nontrivial orbit of which is the image of the nontrivial
ζ-orbit under σ. Hence C σ (:= {ζ σ |σ ∈ SX }) is the cycle decomposition of π σ .
4
i.e., the smallest positive integer such that ζ k = idX
7
1.1.4. Let k ∈ N and m1 , . . . , mk be mutually distinct elements of X, σ ∈ SX . Then
σ
(1) m1 . . . mk = m1 σ . . . mk σ .
(2) m1 . . . mk = m1 m2 m1 m3 · · · m1 mk .
Proof. For all m ∈ X we have (putting mk+1 := m1 , ζ := m1 . . . mk )
(
m,
if m 6∈ {m1 , . . . , mk };
mζ =
mj+1 , if m = mj for some j ∈ k.
First, this equals m m1 m2
it implies, for any σ ∈ SX ,
m1 m3 · · · m1 mk and therefore shows (2). Secondly
mζσ = mσ m1 σ . . . mk σ ,
hence ζσ = σ m1 σ . . . mk σ , i. e., (1).
As a consequence, if ζ = m1 . . . mk is a cycle of length k in SX , then the conjugacy
class ζ SX is the set of all cycles of length k. If n1 , . . . , nk are mutually distinct elements
of X, the bijection mj 7→ nj (j ∈ k) may be extended to a permutation of X, and any
such extension σ has the property that ζ σ = n1 . . . nk , by 1.1.4(1).
A cycle of length 2 is called a transposition. From 1.1.4(2) and 1.1.2 we obtain the
well-known fact that the set of all transpositions is a set of generators of the group
SX . Recall that there exists a (unique) homomorphism of SX into {1, −1} such that
every transposition is mapped to −1, called the sign homomorphism5 . A permutation,
written as a product of transpositions, is an element of the kernel of sgn if and only if
the number of factors is even, and therefore then also called an even permutation. The
subgroup AX := ker sgn of SX is called the alternating group on X and is of index 2 in SX
if |X| > 1. It is obviously generated by all products
τ τ ′ where
τ, τ ′ are transpositions.
If
′
′
′
′
m, n, n are 3 distinct elements of X, then m n m n = m n n . If m,m , n, n′
are 4 distinct elements of X, then m n m′ n′ = m m′ n′ m m′ n . Hence
a product of two distinct transpositions is either a 3-cycle or a product of two 3-cycles.
Summarizing, we obtain
1.1.5. The set of all transpositions is a generating set of SX , the set of all 3-cycles a
generating set of AX .
Let π ∈ SX r ,6 ζ an element of the cycle decomposition C of π, m ∈ X such that
mζ 6= m. If |X| ≥ 4, we choose n ∈ X different from mζ −1 , m, mζ and put σ :=
mζ −1 m n . By 1.1.4(1), ζ σ and ζ are distinct but not disjoint as mζ σ = mσ = n 6=
mζ. It follows that ζ σ 6∈ C so that π σ 6= π. This shows:
Q
Assuming that X = n for some n ∈ N, it suffices to put π sgn := i<j∈n jπ−iπ
j−i for all π ∈ Sn .
6
Given a group G, we write  for its trivial subgroup if there is no danger of confusion.
5
8
1.1.6. If idX 6= π ∈ SX and |X| ≥ 4, there exists a 3-cycle σ ∈ SX such that π σ 6= π. In
particular, Z(SX ) =  = Z(AX ) if |X| ≥ 4.7
The group A5 is of order 60 (= 12 5!) and the group of smallest order which is simple (i. e.,
not of order 1 and without nontrivial proper normal subgroups)8 and not of prime order.
In fact, An is simple for every n ≥ 5. We need a few preparations before we prove this
result. For any subgroup U of SX and m ∈ X we set
StabU (m) := {σ|σ ∈ U, mσ = m},
called the stabilizer of m in U. Obviously, StabU (m) is a subgroup of U, for any m ∈ X.
1.2 Proposition. Let n ∈ N>1 , j ∈ n.
(1) (StabAn (j))π = StabAn (jπ) for all π ∈ Sn ,
S
(2) h j∈n StabAn (j)i = An if n ≥ 4,
(3) StabAn (j) is a maximal subgroup9 of An and isomorphic to An−1 .
Proof. (1) For any π, ρ ∈ Sn we have
ρ ∈ (StabAn (j))π ⇔ ρπ
(2) If n ≥ 4,
S
j∈n
−1
∈ (StabAn (j)) ⇔ jπρπ −1 = j and πρπ −1 ∈ An
⇔ jπρ = jπ and ρ ∈ An ⇔ ρ ∈ StabAn (jπ).
StabAn (j) contains every 3-cycle. Hence the claim, by 1.1.5.
(3) The claim is trivial for n ≤ 3. Let n ≥ 4, X := n r {j} and set
f : StabAn (j) → SX , σ 7→ σ|X .
Then f is a monomorphism and leaves the sign unchanged, hence StabAn (j)f ≤ AX . If
π ∈ AX , we extend π to a permutation σ ∈ StabAn (j) by putting jσ := j. Then σf = π.
Hence
StabAn (j) ∼
= An−1
= StabAn (j)f = AX ∼
as |X| = n − 1. Now let ϕ ∈ An r StabAn (j), i := jϕ. For any k ∈ n r {j} we may
choose l ∈ n r {i, j, k} (as n ≥ 4) and put σ := i k l so that we obtain, by (1),
StabAn (k) = StabAn (jϕσ) = (StabAn (j))ϕσ ⊆ hStabAn (j) ∪ {ϕ}i.
Hence hStabAn (j) ∪ {ϕ}i = An , by (2). The claim follows.
7
Of course, also Z(S3 ) =  while Z(A3 ) = A3 .
A normal subgroup must be a union of conjugacy classes, one of them necessarily being {id5 }. The
other conjugacy classes have orders 12, 12, 15, 20 so that only the union of all conjugacy classes is
a subgroup 6= , by Lagrange’s theorem.
9
A maximal subgroup is a proper subgroup which is not contained in any other proper subgroup, i. e.,
with respect to the relation ⊆, a maximal element of the set of all proper subgroups.
8
9
We remark that 1.2 holds likewise if An is replaced by Sn in its formulation.
1.3 Theorem. The group An is simple for every n ∈ N≥5 .
Proof by induction on n, the case of n = 5 being taken for granted. Let n ∈ N>5 and
suppose An−1 to be simple. Put B := StabAn (n), and let N E An . Then B ∩ N E B ∼
=
An−1 , by 1.2(3), hence by our inductive hypothesis either B ∩ N = B (1st case) or
B ∩ N =  (2nd case).
In the first case we conclude that B < N, as B 5 An by 1.2(1), hence N = An by
1.2(3).
Our aim in the second case is to show that N = . Assuming N 6= , we obtain a
contradiction as follows: We have B < BN ≤ An , hence BN = An by 1.2(3). In
particular, |N| = |An : B| = n. Let X be the normal subset N r  of An . We obtain a
homomorphism of An into SX by restricting the usual conjugation within the group An
to X: Let
f : An → SX , σ 7→ σ̄|X .
Then N is centralized by ker f , hence B ∩ ker f E An . Hence B ∩ ker f =  as B is a
non-normal simple subgroup of An . It follows that
An−1 ∼
= Sn−1 ,
= Bf ≤ SX ∼
=B∼
hence |SX : Bf | = 2 so that Bf is normal and contains every 3-cycle of SX . By 1.1.5,
Bf = AX . Thus any even permutation of the set N which leaves the neutral element
fixed is an automorphism of the group N. It is an easy exercise10 to show that this
implies n < 5, a contradiction.
The alternating groups An , n ≥ 5, thus form a so-called series of finite simple groups.
The most trivial series of this kind is given by the groups Cp of prime order p. A third
series of simple groups is given as follows: Let V be an n-dimensional vector space over a
field K and det the determinant epimorphism of GL(V ) into the multiplicative group K̇
of K, SL(V ) := ker det, P SL(V ) := SL(V )/Z(SL(V )). Then P SL(V ) is simple unless
n = 1 or n = 2, |K| ≤ 3. We will give a sketch of a proof but not its details which may
be found, e. g., in [H], II, §6. It is remarkable, however, that there is a close analogy
between the structure of the proof of the simplicity of An . Therefore we mention the
crucial steps and definitions for the case of P SL(V ) together with the corresponding
parts for the case of An , which will reveal most similar proof procedures.
“By nature”, the group Sn permutes the “set of points” n. Considering now similarly the
set X of all 1-dimensional subspaces of V as a “set of points”, the same statement holds
for the group GL(V ). Instead of a mere analogy we have, more precisely, a reduction
of the latter phenomenon to the “inborn” action of a symmetric group (on its natural
set of points): The elements of GL(V ), originally defined as mappings of V onto V ,
induce mappings of P(V ) onto P(V ), in particular, of X onto X . In this sense, we may
restrict each α ∈ GL(V ) to X . This restriction defines a homomorphism of GL(V ) into
SX . In group theoretic investigations, such “new interpretations” of group elements as
10
Assume n ≥ 5. There are distinct elements α, β ∈ X which are not inverses of each other. Put
γ := αβ and
choose δ∈ X such that
α, β, γ, δ are mutually distinct. By hypothesis, the 3-cycles
α β γ , α β δ , α δ β are automorphisms of N . Hence α = βγ, βδ = γ = δα. It
follows that δβγ = δα = γ, δβ = id, hence γ = id, a contradiction.
10
permutations on certain suitably chosen sets X play an important role as will be seen
on many occasions in the sequel. A transvection of V is a vector space automorphism α
of V for which there exists a maximal subspace W of V such that α induces the identity
endomorphism on W and on V /W .
In Sn
An := ker sgn
Let Dn be the set of all 3-cycles on n.
In GL(V )
SL(V ) := ker det
Let T (V ) be the set of all transvections
of V .
hT (V )i = SL(V ).
T (V ) is a conjugacy class in GL(V ) (for
n ≥ 3 even in SL(V )).
Let X be the set of all 1-dimensional subspaces of V , U ∈ X .
For any T , T ′ ∈ X r {U} there exists
α ∈ GL(V ) such that Uα = U, T α = T ′ .
{α|α ∈ SL(V ), Uα = U} is a maximal
subgroup of SL(V ) if n ≥ 2.
Z(GL(V )) = {µc |c ∈ K̇}
where
µc : V → V , v 7→ cv.
Theorem Except when n = 2, |K| ≤ 3,
every proper normal subgroup of SL(V )
is contained in Z(GL(V )).
Hence
P SL(V ) is simple unless n = 1 or
n = 2, |K| ≤ 3.
hDn i = An .
Dn is a conjugacy class in Sn (for n ≥ 5
even in An ).
Let j ∈ n.
For any k, k ′ ∈ n r {j} there exists
π ∈ Sn such that jπ = j, kπ = k ′ .
StabAn (j) is a maximal subgroup of An
if n ≥ 2.
Z(Sn ) = {idn } if n ≥ 3.
1.3 Theorem An is simple for n ≥ 5.
In a famous paper of 1963, W. Feit and J. G. Thompson [FT] succeeded in proving the
long-standing conjecture by Burnside that every non-cyclic finite simple group is of even
order. This key result put world-wide efforts of group theorists into motion which, after
more than 20 years of work of highest intensity, led in 1983 to a first proclamation (by
D. Gorenstein) that a complete classification of all finite simple groups had been achieved.
But a readable presentation of the proclaimed classification was completely out of reach,
and indeed a number of – even very serious – gaps were discovered in the sequel. More
than 25 years of hard work after the first and premature announcement passed until the
experts of the area had completed a number of missing steps. A general conviction is
generally shared now that the classification is correct.11 But with several thousands of
pages of highly specialized lines of reasoning, the classification theorem cannot really be
called accessible in its current state. Nobody can exclude that in a complicated theory
of this length there might have been overlooked details of underestimated depth. On
the contrary, it would be most surprising if this were not the case. The classification
consists of a certain number (most authors count 18) of countably-infinite series of finite
11
R. A. Wilson describes the state of affairs by the words: “The likelihood of catastrophic errors is much
reduced, though not completely eliminated” ([W], 1.4)
11
simple groups and a list of 26 so-called sporadic finite simple groups. Five of the latter,
the so-called Mathieu groups, had been discovered already between 1860 and 1870. Not
before 1964, a further sporadic group was discovered by Z. Janko (the first Janko group
J1 of order 175560). It may be defined as the subgroup of GL(7, 11) generated by the
two elements




−3
2 −1 −1 −3 −1 −3
0 1 0 0 0 0 0
−2
 0 0 1 0 0 0 0
1
1
3
1
3
3




−1 −1 −3 −1 −3 −3
 0 0 0 1 0 0 0
2




.
0 0 0 0 1 0 0 , −1 −3 −1 −3 −3
2
−1




−3 −1 −3 −3
 0 0 0 0 0 1 0
2 −1 −1




 1
 0 0 0 0 0 0 1
3
3 −2
1
1
3
3
3 −2
1
1
3
1
1 0 0 0 0 0 0
The sporadic group of largest order, often called the Fischer-Griess monster group or
friendly giant, contains more than 8 · 1053 elements and arises as the automorphism
group of some non-associative commutative algebra of dimension 196882 over the field
of 2 elements. The prime divisors of its order are all the primes between 2 and 31, plus
41, 47, 59, 71. It is known to involve all other sporadic simple groups as factor groups
of subgroups, with 6 exceptions, the so-called pariahs. One of these exceptions is the
group J1 .
We leave the fascinating topic of the classification project the final version of which is
certainly still far from being discovered. The highly specialized techniques developed in
its pursuit are not the aim of this course but will become accessible on the grounds of the
ideas which are presented here. The interested reader may study the last three chapters
of [KS] as an introduction into that area. A description of all finite simple groups (under
the assumption that the current classification result is correct) is the ambitious object
of [W].
1.4 Definition. An operating system is a triple (Ω, X, f ) where Ω, X are sets and f is a
mapping of Ω into X X . The mapping f is called an action of Ω on X, and the elements
of Ω are called operators on X (with respect to f ). In many cases it may be assumed
that the operators are mappings from X into X right from the beginning, i. e., f = id.
There are two very different group-theoretic specializations, both of utmost importance.
Let G be a group.
1) G is passive: X = G. Let (Ω, G, f ) be an operating system where f is a mapping of
Ω in End G. Then G is called a group with set of operators Ω.
2) G is active: Ω = G. Let (G, X, f ) be an operating system where f is a homomorphism
of G into SX . Then f is called a group action of G on X.
We will first consider type 2) of an operating system. The group action f is called faithful
if f is injective, i. e., if ker f = . If there is no danger of confusion (in particular, if
there is a unique group action f considered), we write simply xg instead of x(gf ) (where
x ∈ X, g ∈ G). Furthermore, for every x ∈ X the set
xG := {y|y ∈ X, ∃g ∈ G xg = y}
12
is called the G-orbit of x. The group action is called transitive if xG = X for all x ∈ X.
1.4.1. If G acts on a set X, then the relation ∼ given by
G
x ∼ y ↔ ∃g ∈ G xg = y
G
(x, y ∈ X)
is an equivalence on X, and the equivalence classes are exactly the G-orbits in X. In
particular, the G-orbits form a partition of X.
This partition is usually denoted by X/∼ or X/G. There are two standard actions of G,
G
both with Ω = G = X:
(a) f : G → SG , g 7→ ḡ,
i. e., G acts on G via conjugation. Then ker f = Z(G) (see 1.0.1), f is faithful if and
only if Z(G) = . Conjugation is written exponentially. Accordingly, the orbit of an
element x ∈ G under conjugation, i. e., the conjugacy class of x, is denoted by xG (cf.
p. 6).
(b) ρ : G → SG , g 7→ ρg (where ρg : G → G, x 7→ xg),
i. e., G acts on G via right multiplication. Then ker ρ = , i. e., ρ is faithful. More
precisely, ρ is elementwise fixed-point-free, which means that the set Fix g of all fixed
points of g ∈ G is empty unless g = 1G . Furthermore, xG = {xg|g ∈ G} = G, i. e., ρ is
transitive. As ρ is a monomorphism, we conclude
1.4.2. (Cayley’s theorem) Every group G is isomorphic to a subgroup of SG .
Left multiplication does not give a homomorphism, but an anti-homomorphism of G
into SG . This implies that the inverse left multiplication λ : G → SG , g 7→ λg−1 (where
λh : G → G, x 7→ hx), is a transitive elementwise fixed-point-free action of G on G. We
have the following connection between the standard actions of G:
1.4.3. ∀g ∈ G (gλ)(gρ) = ḡ = (gρ)(gλ).
1.4.4. If f is a group action of G on a set X und H ≤ G, then f|H is a group action of
H on X.
With respect to ρ, λ we have: The H-orbits in G regarding ρ|H (λ|H resp.) are the left
cosets xH (the right cosets Hx) where x ∈ G. We write G /. H (G ./ H resp.) for the
set of all right cosets (left cosets) of H in G.
G /. H → G /. H
is a transitive
1.4.5. Let H ≤ G. Then fH : G → SG /. H , g 7→
Hx 7→ Hxg
T
action of G on G /. H, ker fH = x∈G H x .12
Proof. T
g ∈ ker fH ⇔ ∀x ∈ G Hxg = Hx ⇔ ∀x ∈ G xgx−1 ∈ H ⇔ ∀x ∈ G g ∈ H x
⇔ g ∈ x∈G H x for all g ∈ G.
12
The same holds for the inverse left multiplication on G ./ H.
13
T
The subgroup x∈G H x is the largest normal subgroup of G which is contained in H,
called the core of H in G and commonly denoted by HG . Recall that any group homomorphism f of G induces a group homomorphism fG/N of a quotient G/N whenever
N E G such that N ≤ ker f . In particular, this holds for group actions f . Thus if
HG ⊇ N E G then G/N acts on G /. H via right multiplication. As a first application of
1.4.5, we let G be the alternating group and prove the following supplement of 1.2(3):
1.4.6. Let n ∈ N, H a subgroup of index n of An . Then H ∼
= An−1.
Proof. For n ≤ 3 the claim is trivial. If n = 4, we have |H| = 3, hence H ∼
= A3 . Let
n ≥ 5, X := An /. H. By 1.3, HAn =  so that fH is a monomorphism of An into SX .
As  6= AX ∩ An fH E An fH ∼
= AX , it follows that An fH = AX . We restrict fH to
H and thus obtain an action fH′ of H on the set X ′ := X r {H} because the element
H ∈ X is fixed by H. We have ker fH′ = H ∩ ker fH = . As An fH = AX , HfH′
consists of even permutations of X ′ so that fH′ is a monomorphism of H into AX ′ . But
|H| = n1 |An | = |AX ′ |, hence H ∼
= An−1.
= AX ′ ∼
Let (Ω, X, f ) be an operating system. For all ω ∈ Ω we put
FixX,f (ω) := {x|x ∈ X, x(ωf ) = x},
and for all x ∈ X
StabΩ,f (x) := {ω|ω ∈ Ω, x(ωf ) = x},
in short FixX (ω), StabΩ (x) resp.,13 if there is no doubt about the action f . For all n ∈ N
we have a natural “componentwise” action of Ω on X n , given by
Xn
→
Xn
(n)
n Xn
,
f : Ω → (X ) , ω 7→
(x1 , . . . , xn ) 7→ (x1 (ωf ), . . . , xn (ωf ))
and an action on P(X) by
⊂
f : Ω → P(X)
P(X)
P(X) → P(X)
, ω 7→
T
7→ T (ωf )
where T (ωf ) := {t(ωf )|t ∈ T }. For all T ⊆ X
NΩ,f (T ) := Stab
⊂
Ω, f
(T )
is called the normalizer,
\
CΩ,f (T ) :=
StabΩ,f (x)
= {ω|ω ∈ Ω, T (ωf ) = T }
= {ω|ω ∈ Ω, ∀x ∈ T
x∈T
x(ωf ) = x}
the centralizer of T in Ω with respect to f . Again, we write NΩ (T ), CΩ (T ) resp., if the
reference to f is obvious.
13
Many authors use the notation CX (ω), CΩ (x) resp. the latter of which is also in concordance with
what we define later for subsets T in place of elements x of X.
14
We now consider again the special situation of an action of a group G: Let f be a
⊂
homomorphism of G into SX for some set X. Then f is a homomorphism of G into
SP(X) . Since gf is injective for every g ∈ G, we have |T | = |T (gf )| for every T ⊆ X.
Setting Pl (X) := {T |T ⊆ X, |T | = l} for every l ∈ N, this implies
⊂
1.4.7. For all l ∈ N, f induces an action of G on Pl (X).
Obviously, StabG,f (x) is a subgroup of G, for every x ∈ X. Applying this to the action
⊂
f , we conclude that the same holds for NG,f (T ) for every T ⊆ X. If T ⊆ X, f induces
an action of NG,f (T ) on T the kernel of which is CG,f (T ). Hence we have
1.4.8. CG,f (T ) E NG,f (T ) for all T ⊆ X.
From 1.4.1 we obtain
Ṡ
1.4.9. For all T ⊆ X, T = x xNG,f (T ) where x ranges over a set of representatives of
the orbits of NG,f (T ) in T .14
If there is no reference to some special set X and action f of a group G, then it is tacitly
assumed that X = G and the action is the 1st standard action (conjugation), i. e., for
any T, U ⊆ G,
NU (T ) = {g|g ∈ U, T g = T },
CU (T ) = {g|g ∈ U, ∀x ∈ T
xg = x}.
For arbitrary group elements g, x we have xg = x if and only if g x = g. Therefore CU (T )
may be interpreted as the set of all g ∈ U fixing T elementwise and likewise as the set of
all g ∈ U which are fixed points under the action of T . If T is a subgroup of G, then the
permutations of the set T induced by NG (T ) by conjugation are in fact automorphisms
of the group T . It follows that NG (T )/CG (T ) is then not only a subgroup of ST but,
more restrictively, of Aut T :
1.4.10. If T ≤ G, then NG (T )/CG (T ) is isomorphic to a subgroup of Aut T .
1.5 Definition. Let Ω be a set acting on sets X, Y . A mapping ϕ of X into Y is then
called Ω-compatible if
∀x ∈ X ∀ω ∈ Ω (xω)ϕ = (xϕ)ω.
If there exists a bijective Ω-compatible mapping of X onto Y , the X, Y are called Ωequivalent or Ω-similar, written X ≈ Y . We illustrate these actions considering certain
actions of a group G:
Ω
1.5.1 Example. Let G act upon a set X via f , x ∈ X, H := StabG (x). If a, b ∈ G such
that Ha = Hb, it follows that b = ha for some h ∈ H, hence x(bf ) = x(hf )(af ) = x(af ).
14
Ṡ
denotes a disjoint union.
15
We put ϕ : G /. H → X, Ha 7→ x(af ) (which is well-defined as we have just seen). For
all a, g ∈ G we have
((Ha)g)ϕ = x(ag)f = x(af )(gf ) = ((Ha)ϕ)(gf ).
Furthermore, ϕ is injective: (Ha)ϕ = (Hb)ϕ ⇒ x(af ) = x(bf ) ⇒ ab−1 ∈ H ⇒ Ha =
Hb, for every a, b ∈ G. Clearly, ϕ is surjective if and only if the action of G on X is
transitive. Hence in this case X and G /. H are G-equivalent. We thus have
1.5.2. Up to G-equivalence, the transitive actions of G are those given by 1.4.5. An
action of G on a set X is transitive if and only if X ≈ G /. H for a subgroup H of G.
G
1.5.3 Example. Let f be a group action of G on a set X, N E G, Y := X/N. Then Y
⊂
is G-invariant (with respect to f ) as
∀x ∈ X ∀g ∈ G (xN)g = (xgg −1 )Ng = (xg)N.
(∗)
⊂
Therefore, f induces a group action of G on Y , and (∗) shows that
ϕ : X → Y, x 7→ xN,
is G-compatible. While ϕ is obviously surjective, it is injective if and only if xN = {x}
for all x ∈ X, i. e., if and only if N ⊆ ker f .
The hypotheses of 1.5.3 are satisfied, for example, if two groups H, M act on the same
set X (via fH : H → SX , fM : M → SX ) in such a way that HfH ≤ NSX (MfM ). We
put G := NSX (MfM ) and apply 1.5.3 with respect to the normal subgroup MfM of G.
Then we have:
1.5.4. Let H, M be groups acting on a set X via group actions fH , fM resp., such that
HfH ≤ NSX (MfM ). Then fH induces an action of H on X/M, and the mapping
ϕ : X → X/M, x 7→ x(MfM ),
is H-compatible.
We say that the action of M is normalized by the action of H if the hypotheses of 1.5.4
are satisfied. This simple concept of linking two group actions, sketched by the following
diagram, will turn out to be of fundamental importance.
NSX (M fM )
M
-
M fM
HfH

16
H
An even more restrictive condition is that the action of M be centralized by the action of
H, i. e., HfH ≤ CSX (MfM ). A simple example is provided by 1.4.5: Let H ≤ G =: X.
The group actions
ρ : G → SG , g 7→ gρg ,
λ|H : H → SG , , h 7→ λh−1
centralize each other (Gρ, Hλ commute elementwise). Furthermore, G /. H is the set
of H-orbits in G. The action on G /. H induced by ρ is given by Hx 7→ Hxg (for any
g ∈ G), and
ϕ : G → G /. H, x 7→ Hx,
is G-compatible.
1.6 Proposition (on transitive group actions). Let G be a group which acts transitively
on a set X. Let x ∈ X.
(1) (General Frattini argument) ∀H ⊆ G
xH = X ⇔ StabG (x)H = G.
(2) |X| = |G : StabG (x)|.
Proof. (1) Let H ⊆ G. If xH = X, g ∈ G, we have xg ∈ xH, hence xg = xh for some
h ∈ H, i. e., gh−1 ∈ StabG (x), g ∈ StabG (x)H. Conversely, if StabG (x)H = G, then
X = xG = xStabG (x)H = xH.
(2) follows from 1.5.1.
1.7 Corollary. Let G be a finite group acting on a finite set X, B := X/G.
P
(1) (class equation of a group action) |X| = B∈B |B|,
(2) ∀B ∈ B ∀x ∈ B
|B||StabG (x)| = |G|,
|xG ||CG (x)| = |G|,
P
1
(3) (Cauchy-Frobenius Lemma) |B| = |G|
g∈G |FixX g|
in particular: ∀x ∈ G
15
Proof. (1) Obvious by 1.4.1. (Here the finiteness of G is not needed.)
(2) The action of G on an orbit is transitive by definition. Hence the claim follows from
1.6(2). The final assertion is the special case where G acts on G by conjugation.
P
P
P
P
1
1
(3) We have, by 1.4.1, x∈X |xG|
= B∈B x∈B |B|
= B∈B 1 = |B|. Hence, applying
1.6(2) in the third step,
X
g∈G
15
|FixX g| =
X
1=
(g,x)∈G×X
xg=x
X
|StabG (x)| =
x∈X
X |G|
= |G||B|.
|xG|
x∈X
The equation may be interpreted as follows: The number of orbits equals the “average number of
fixed points”.
17
Under appropriate hypotheses on the orders of the involved groups we will now obtain
a number of statements of fundamental importance in group theory. Recall that a finite
group the order of which is a power of a prime p is called a p-group. A p′ -group is a finite
group the order of which is not divisible by p. These may be viewed as special cases
of the notion of a π-group for a set π of prime numbers, which is a group the order of
which has only prime divisors in π. The set of all primes 6∈ π is denoted by π ′ .
1.8 Proposition. Let G be a group, p a prime.
(1) If H is a p-subgroup of G, T a finite H-invariant16 subset of G such that p ∤ |T |,
then CT (H) 6= ∅.
(2) Let G be a p-group. If  6= N E G, then N ∩ Z(G) 6= .
In particular, if G 6= , then Z(G) 6= . Moreover, G is soluble.
(3) Let H be a p-subgroup of G. If T is an H-invariant coset of some p′ -subgroup of G,
then CT (H) 6= ∅.
In particular, if N is a normal p′ -subgroup of G, then CG/N (H) = NCG (H)/N.
Proof. (1) If CH (T ) = ∅, then the order of each H-orbit in T is a power 6= 1 of p, by
1.7(2). Hence |T |, being a sum of powers 6= 1 of p by 1.7(1), is divisible by p.
(2) For the first assertion we set T := N r , H := G in (1). The second assertion is the
special case N = G. The third now follows by induction on |G|, applying the inductive
hypothesis to G/Z(G) which is a p-group of smaller order than |G| if G 6= .
(3) By hypothesis, |T | is not divisible by p so that the first statement is a special case
of (1). Under the hypothesis of the final assertion we may apply this to every coset of
N in G and obtain the claim.
1.8.1. If the order of a group G is the square of a prime, then G is abelian.
Proof. By 1.8(2), Z(G) 6=  so that G/Z(G) is cyclic. The claim follows from 1.0.2.
1.8.2. Let p be a prime and G a p-group, H < G. Then H ⊳ G and |G : H| = p. In
max
particular, G′ ≤ H.
Proof by induction on |G|. If |G| = 1, there is nothing to prove. Let |G| > 1 and assume
the claim holds for p-groups of smaller order. By 1.8(2), Z(G) 6= , hence there exists
a central subgroup C of order p. If C ≤ H, we have H/C < G/C, inductively H/C
max
is normal and of index p in G/C so that the claim follows. Otherwise H < HC ≤ G,
hence HC = G as H < G. This implies H ⊳ G because C ≤ Z(G) and |G : H| = p
max
because |C| = p. This completes the induction. G/H is abelian, hence G′ ≤ H.
16
with respect to conjugation
18
In these applications of 1.7 we considered the 1st standard action, the conjugation. In
the following we shall work with the 2nd standard action, the right multiplication. A
useful first remark generalizes the well-known assertion that a subgroup of index 2 is
always normal:
1.8.3. Let G be a finite group and H < G. If |G : H| is the smallest divisor 6= 1 of |G|,
then H ⊳ G.17
Proof.
H acts by right multiplication on G /. H r {H}. Let B be an H-orbit. By 1.7(2),
|B| |H| |G|. Now |B| < |G : H| implies |B| = 1. Thus HxH = Hx, i. e., xH = Hx,
for all x ∈ G r H, hence also for all x ∈ G.
1.9 Proposition. Let G be a group and k, l ∈ N such that |G| = kl. Let Ul (G) be the
set of all subgroups of order l of G.18 We consider the action of G on Pl (G) given by
right multiplication (see 1.4.7) and set B := Pl (G)/G, B(k) := {B|B ∈ B, |B| = k}.
(1) k |B| for all B ∈ B.
(2) If B ∈ B(k) , then B = G /. H for some H ∈ Ul (G).
(3) |Ul (G)| = |B(k) |.
(4) If l is a power of a prime p, then |Ul (G)| ≡ 1. In particular, Ul (G) 6= ∅.
p
The proof of 1.9 will yield in passing the following two elementary number-theoretic
remarks:
1.9.1. Let k, l ∈ N. Then
(1) k| kll .
(2) If l is a power of a prime p, then
1 kl
k l
≡ 1.
p
Proof of 1.9. (1), (2) Let f be the action of G on G given by right multiplication, B ∈ B,
T ∈ B, GT := NG,f (T ) = Stab ⊂ (T ). Since |xGT | = |GT | for all x ∈ G we have, by
G, f
1.4.9,
|G|
.
|GT ||T | = l =
k
Hence k |G| = |B| by 1.6(2).19 If |B| = k we have |GT | = |G| = |G| = l = |T | so that
|GT |
|B|
1.4.9 implies T = xGT for an x ∈ T . Now let H :=
−1
GTx .
k
It follows that
B = {T g|g ∈ G} = {xGT g|g ∈ G} = {H(xg)|g ∈ G} = G /. H.
17
This shows, in particular, that the first claim in 1.8.2 (H ⊳ G) is a consequence of the second
(|G : H| = p).
18
Recall that |Ul (G)| = 1 if G is cyclic. – In general it may happen that Ul (G) = ∅, e. g., if l = 6,
G = A4 .
P
19
By 1.7(1) it follows that k B∈B |B| = |Pl (G)| = kll . Clearly, a group of order kl exists (at least
the cyclic one) so that we obtain 1.9.1(1).
19
(3) The mapping ψ : Ul (G) → B(k) , H 7→ G /. H is surjective by (2). Being the only
element of G /. H which is a subgroup, we recover H from G /. H whence ψ is injective.
s
|B||G| = kps for all B ∈ B.
(4) Let l =
p
for
some
s
∈
N
.
By
(1)
and
1.7(2)
we
have
k
0
Hence pk |B| unless |B| = k. It follows that
X
X
X
kl
= |Pl (G)| =
|B| +
|B| = k|Ul (G)|,
|B| ≡
pk
l
B∈B
(k)
(k)
B∈B
hence
|B|6=k
B∈B
1 kl
≡ |Ul (G)|.
p
k l
(∗)
As this holds so far for an arbitrary group of order kl, we may apply (∗), in particular,
to the cyclic group of order kl for which we know that the right-hand side equals 1. This
proves 1.9.1(2). By means of this congruence, (∗) now implies |Ul (G)| ≡ 1.
p
1.10 Definition.
Let G be a finite group, p a prime and t ∈ N0 maximal with the
property that pt |G|. a Sylow p-subgroup of G is a subgroup P of G such that |P | = pt .
We set
[
Sylp (G).
Sylp (G) := {P |P ≤ G, |P | = pt }, Syl(G) :=
p||G|
p prime
Note that  6∈ Syl(G) but Sylp (G) =  for all primes p which do not divide |G|.
1.11 Theorem (Sylow 1872). Let G be a finite group, p a prime.
(1) |Sylp (G)| ≡ 1, in particular: Sylp (G) 6= ∅.
p
(2) If H is a p-subgroup of G and P ∈ Sylp (G), there exists an element g ∈ G such that
H ≤ P g.
(3) Sylp (G) is a class of conjugate subgroups of G, and
∀P ∈ Sylp (G) |Sylp (G)| =
|G|
.
|NG (P )|
Proof. (1) is a special case of 1.9(4).
(2) H acts on G /. P via right multiplication. Let B := (G /. P )/H. By 1.7(1),
X
p ∤ |G : P | =
|B|
B∈B
and
∀B ∈ B
|B||H|,
by 1.7(2). Hence there exists an H-orbit of length 1, i. e., there exists an element g ∈ G
such that (P g)H = P g. It follows that P g H = P g , hence H ≤ P g .
20
(3) If H, P ∈ Sylp (G), there exists an element g ∈ G such that H ≤ P g , by (2). Since
|H| = |P | = |P g |, it follows that H = P g . Thus G acts transitively on Sylp (G) by
conjugation. By 1.6(1), this implies |Sylp (G)| = |G : NG (P )|.
The idea of using the line of reasoning in 1.9(4) to prove 1.11(1) is due to H. Wielandt.
Of course, 1.9.1 may be proved independently of this group-theoretic context, and some
authors indeed use its assertions as auxiliary statements borrowed from elementary number theory to prove the existence part of Sylow’s theorem following Wielandt. But it
is a benefit of the proof that it simultaneously yields the remarks 1.9.1(1),(2), avoiding
any separate number-theoretic consideration. It should be emphasized that this is one
of many examples where some simple algebraic argument has a number-theoretic consequence – for free. Making use of the cyclic group in the proof is therefore in no way a
detail which should be replaced by some other line of reasoning. On the contrary: An
analysis of the way in which 1.9.1(2) was obtained may even suggest that Wielandt’s
idea can be viewed as an ingenious reduction of the case of an arbitrary finite group to
that of the cyclic group of the same order.
Conceptually, Wielandt’s proof of 1.11(1) is extremely simple. Omitting the routine
technical steps involved, in the view of a group-theoretic expert it boils down to just
three lines (see [W], 1.5). Still it should be noted that 1.9(3) holds without any restriction on the divisor l of |G|.
1.12 Corollary. (Classical Frattini argument) Let G be a group, M a finite normal
subgroup of G, P ∈ Syl(M). Then NG (P )M = G.
finite
M

P

G
NG (P )
Proof. Let p be the prime such that P ∈ Sylp (M). By 1.11(3), M,
a fortiori G, acts transitively by conjugation on Sylp (M), i. e., P M =
Sylp (M) = P G . Applying 1.6(1) (where X = Sylp (M), x = P , H =
M), we obtain NG (P )M = G.
By 1.11(1),(3), the index of NG (P ) in G must be both
a divisor of |G : P | and congruent 1 modulo p
if P is a Sylow p-subgroup of G. There are many examples where these two conditions
are extremely restrictive, hence can provide some most important structural information
on G. We give a few illustrations.
1.12.1 Example. We determine the groups of order 1225. Let G be a group of order
1225 = 52 72 , G5 ∈ Syl5 (G), G7 ∈ Syl7 (G). Then G5 ∩G7 = , hence |G5 G7 | = |G5 ||G7| =
|G| so that G5 G7 = G. Now |G : G5 | = 72 and has the divisors 1, 7, 72 . The only divisor
which is congruent 1 modulo 7 is 1. It follows that NG (G7 ) = G, i. e.,
G
G7 E G. Similarly, |G : G7 | = 52 and has the divisors 1, 5, 52 . The only
G5
G7 divisor which is congruent 1 modulo 5 is 1. It follows that NG (G5 ) = G,
i. e., G5 E G. We conclude that G ∼
= G5 × G7 . The groups G5 , G7 have

orders 25, 49 resp., hence are abelian by 1.8.1. In G5 , either there is an element of order 25 (and G5 cyclic in this case) or all non-trivial elements are of order 5
21
(and G5 ∼
= C5 × C5 in this case). The same holds analogously for G7 so that we obtain that G must be, up to isomorphism, one of the (mutually non-isomorphic) abelian
groups
C52 × C72 , C5 × C5 × C72 , C52 × C7 × C7 , C5 × C5 × C7 × C7 .
1.12.2. Let G be a group of order 30. Then |Syl3 (G)| = 1 = |Syl5 (G)|.
Proof. We exploit repeatedly the arithmetic condition mentioned before 1.12.1. The
only divisor 6= 1 of 30 which is congruent 1 modulo p is 10 for p = 3, 6 for p = 5.
Therefore |Syl3 (G)| =
6 1 6= |Syl5 (G)| would imply that G has 20 elements of order 3 and
24 elements of order 5 which is absurd. Hence G has a normal Sylow 3-subgroup G3
or a normal Sylow 5-subgroup G5 . But then the 15 elements of G3 G5 form a subgroup.
Both Sylow subgroups of this subgroup are normal in G3 G5 (as the only divisors of 15
are 1, 3, 5, 15). But then NG (G3 ) and NG (G5 ) are of order ≥ 15, i. e., of index ≤ 2 in
G. This implies that NG (G3 ) = G = NG (G5 ), hence the claim.
1.12.3. Let G be a group of order 60 such that |Syl5 (G)| 6= 1. Then G ∼
= A5 . In
particular, every simple group of order 60 is isomorphic to A5 .
Proof. We first observe that G has no subgroup of index 2: Such a subgroup would
have a unique Sylow 5-subgroup, by 1.12.2. The index of its normalizer in G would
therefore be ≤ 2, hence = 1 as this index must be ≡ 1 modulo 5. This contradicts
our hypothesis. Moreover, G has no normal subgroup of order 2 because its quotient
would have a subgroup of order 15, by 1.12.2, i. e., of index 2. But then G would have
a subgroup of index 2 which, as we have seen, is not the case.
Now let G5 ∈ Syl5 (G). Then 1 ≡ |Syl5 (G)| = |G : NG (G5 )| 12 so that our hypothesis
5
implies that |Syl5 (G)| = 6, |NG (G5 )| = 10. The action of G on G /. NG (G5 ) by right
multiplication induces a homomorphism f of G into S6 the kernel of which is a subgroup
of NG (G5 ). We know that | ker f | =
6 2, and ker f = G5 or ker f = NG (G5 ) would imply
that G5 E G which is absurd. It follows that ker f = . Hence f is a monomorphism
of G into S6 . Since Gf ∩ A6 is of index ≤ 2 in Gf and G has no subgroup of index
2, it follows that Gf ≤ A6 . Thus Gf is a subgroup of index 6 in A6 . It follows that
G∼
= Gf ∼
= A5 , by 1.4.6.
22
2 Groups with operators
We shall now consider operating systems of type 1) in 1.4. Natural sets of operators on
a group G are
1a) Ω = End G,
1b) Ω = Aut G,
1c) Ω = G
where the action f of Ω on G is given by idΩ in 1a), 1b), and by f : G → Aut G, g 7→ ḡ
(conjugation) in 1c). An End G-invariant subset of G with respect to conjugation is called
fully invariant, an Aut G-invariant subset characteristic, and a G-invariant subset of G is
called normal (cf. p. 6) in G. We write H ≤ G to say that H is a characteristic subgroup
char
of G. It suffices to consider an arbitrary non-trivial proper subgroup of G = Cp × Cp
(p a prime) to see that normal subgroups need not be characteristic in G. Z(G) is
characteristic but generally not fully invariant in G: If G = C2 × S3 , then Z(G) is the
(unique) direct factor of order 2 of G, and there is an endomorphism of G which maps
Z(G) onto a non-central subgroup of order 2 of G.
The commutator subgroup G′ = h[x, y]|x, y ∈ Gi 20 of a group G is fully invariant, a
fortiori characteristic in G, because [x, y]α = [xα , y α] ∈ G′ for all x, y ∈ G, α ∈ End G.
For arbitrary subsets U, V of G we set
[U, V ] := h[u, v]|u ∈ U, v ∈ V i
so that G′ = [G, G]. Recall the following characterization of G′ as the smallest normal
subgroup of G with abelian factor group.
2.0.4. Let H ≤ G. Then H E G and G/H is abelian if and only if G′ ≤ H.
Every vector space V over a field K is an example of an (abelian) group with operator set
K, where the action of K on V is given by associating to each scalar c the multiplication
by c. More generally, this holds if K is a commutative unitary ring and we have a
multiplication between K and an abelian group V satisfying the conditions known from
the classical vector space axioms. In this case we call V a K-space. A K-representation of
a K-space A is a K-linear mapping of A into the K-space EndK V of all endomorphisms
of a K-space V which is then called an A-module over K. In particular, every K-space
is a K-module over K. If A is a unitary associative algebra, a representation of A which
20
Recall that [x, y] := x−1 y −1 xy = x−1 xy , [x, y] = [y, x]−1 . Replacing the conjugation by y by an
arbitrary group endomorphism γ, one defines, more generally, [x, γ] := x−1 xγ .
23
is a multiplicative homomorphism of A is called an algebra representation of A and unital
if 1A acts as the identity on V . Clearly, every A-module V over K is an example of a
group with operators.
2.1 Definition. Let G be a group and Ω a set of operators on G via a mapping f of
Ω into End G. The notation T ⊆ G means that T is an Ω-invariant subset of G, i. e.,
Ω
T ⊆ G and T α ⊆ T for all α ∈ Ωf . If N E G, then Ω is also a set of operators of N
Ω
and of G/N. G is called Ω-simple if G 6=  and , G are the only Ω-invariant normal
subgroups of G. A direct Ω-factor of G is an Ω-invariant normal subgroup N of G for
21
˙
which there exists an Ω-invariant normal subgroup M such that G = N ×M
. The fact
that this equation is satisfied for some Ω-invariant normal subgroups M, N is expressed
˙ M. If G 6=  and G allows only the trivial possibilities that
by the notation G = N ×
Ω
N =  or M =  then G is called directly Ω-indecomposable. A direct Ω-decomposition of
G is a set X of Ω-invariant normal subgroups 6=  of G such that
Y
Y
X = G,
∀N ∈ X
(X r {N}) ∩ N = .
If G = , then ∅ is a direct Ω-decomposition of G. If G 6= , then {G} is a direct
Ω-decomposition of G.
If Ω also acts on a further group G∗ , then an Ω-compatible group homomorphism of G
into G∗ is called an Ω-homomorphism. From this we obtain the notions Ω-monomorphism,
Ω-epimorphism, Ω-isomorphism, Ω-endomorphism, Ω-automorphism in the obvious way and
write EndΩ G (AutΩ G resp.) for the set of all Ω-endomorphisms (Ω-automorphisms resp.)
of G. We write G ∼
= G∗ if there exists an Ω-isomorphism of G onto G∗ . One readily
Ω
verifies the homomorphism theorem for groups with sets of operators (which will be used
without further reference): The image of G under an Ω-homomorphism ϕ into G∗ is an
Ω-subgroup of G∗ , the kernel is a normal Ω-subgroup of G, and Gϕ ∼
= G/ ker ϕ.
Ω
An important example arises with any bipartite Ω-decomposition of G, i. e., a pair π =
(U, V ) of (not necessarily normal) Ω-subgroups of G such that UV = G, U ∩ V = .
The corresponding projections πU , πV of G onto U, V resp., are defined by the condition
that gπU is the unique element u ∈ U such that g ∈ uV , gπV is the unique element
v ∈ V such that g ∈ Uv, for every g ∈ G. An Ω-factor of G is a component of a bipartite
Ω-decomposition of G. If V is an Ω-factor of G, every U ≤ G such that (U, V ) is an
Ω-decomposition of G is called an Ω-complement of V in G.
Ω
2.1.1. If π = (U, V ) is a bipartite Ω-decomposition of G, then πU , πV are Ω-compatible
mappings. If V E G, then πU is an Ω-epimorphism of G onto U and ker πU = V .
Proof. If g = uv where u ∈ U, v ∈ V , then g α = uα v α and uα ∈ U, v α ∈ V , for all
α ∈ Ωf . If V EG, u, ũ ∈ U, v, ṽ ∈ V , then uv ũṽ = uũ v ũ ṽ, hence πU is a homomorphism,
and clearly surjective.
21
i. e., G = N M and N ∩ M =  so that G is isomorphic to the direct product N × M .
24
If (U, V ) is a bipartite Ω-decomposition of G and V EG, then (U, V ) is called a semidirect
˙ V . The symbol Ω is omitted
Ω-decomposition of G which is expressed in the form G = U ⋉
in a context where no such set plays a role.
Ω
2.1.2. Let π := (U, V ) be a semidirect decomposition of G and T E G such that πV
induces a bijection of T onto V . Then ψ := (U, T ) is a semidirect decomposition of G,
πV |T ψT = idT , ψT |V πV = idV .
Proof. U ∩ T =  as πV |T is injective. For any g ∈ G there exists an element x ∈ T
such that gπV = xπV , therefore g, x ∈ Uv for some v ∈ V . It follows that gx−1 ∈ U,
g ∈ UT . The two final assertions follow from the trivial equivalence v = ut ⇔ t = u−1 v
(t ∈ T, u ∈ U, v ∈ V ).
2.2 Proposition. Let G be a group, Ω a set of operators on G. Let π := (U, V ) be a
semidirect Ω-decomposition of G and T E G such that πV induces a bijection of T onto
Ω
V , ψ := (U, T ). Suppose T ≤ UG V and set
γ : G → G, g 7→ g πU g ψT .
G
V
T
U

UG
Then γ ∈ AutΩ (G), g −1g γ = g ψT πU for all g ∈ G. In particular,
[G, γ] = T πU ≤ Z(UG ),
[G, γ, γ] ≤ [U, γ] = ,
V γ = V ψT = T.
˙ V = U×
˙ T , T πU ≤ Z(G).
Special case: If U E G, then G = U ×
Ω
Ω
Proof. We have uγ = u for all u ∈ U, v γ = v ψT for all v ∈ V directly from the definition
of γ. In particular, U, T ⊆ Gγ . We observe, for any u, ũ ∈ U, v ∈ V , t ∈ T ,
(∗)
uv = ũt ⇒ ∀x ∈ T
xũ = xu ,
because t = ũ−1uv implies that ũ−1 u ∈ UG ≤ CG (T ) as T ≤ UG V . Now u ∈ ũCG (T )
and (∗) follows.
To prove that γ is a homomorphism let g, g ′ ∈ G, u, ũ, u′ , ũ′ ∈ U, v, v ′ ∈ V , t, t′ ∈ T
such that uv = g = ũt, u′ v ′ = g ′ = ũ′t′ . Then u′ uv ′u v = g ′ g = ũ′ ũt′ũ t. By means of (∗),
(g ′ g)γ = u′ ut′ũ t = u′ut′u t = u′t′ ut = g ′γ g γ .
As U ∩ T =  we have g ∈ ker γ if and only if g πU = 1G = g ψT , i. e., g ∈ V ∩ U = . We
have shown that γ is an injective endomorphism of G such that U, T ⊆ Gγ . Therefore γ
is an automorphism and Ω-compatible by 2.1.1. Furthermore, [T πU , UG ] = [T, UG ] = .
If g ∈ G and u ∈ U, t ∈ T such that g = ut, then g −1 g γ = t−1 tγ = tπU t = tπU , hence
g −1 g γ = g ψT πU . Finally, γV = ψT |V is an isomorphism of V onto T by 2.1.2.
Moreover we observe that, under the hypotheses of 2.2, CG (γ) = U(V ∩ T ).
25
2.3 Definition. Let G be a group. For any set Y , the set GY of all mappings of Y into
G is a group with respect to the (in general non-commutative) operation ∔ defined by
α∔β :
Y → G, y 7→ (yα)(yβ) for all α, β ∈ GY .
The function ȯ : Y →  is its neutral element. Given α ∈ GY , the function Y → G,
y 7→ (yα)−1, is the inverse of α. We now consider the special case of Y = G. The
set GG is a (multiplicatively written) monoid with respect to the usual composition of
mappings, and we trivially have the left-sided distributive law
2.3.1. ∀γ, α, β ∈ GG
γ(α ∔ β) = γα ∔ γβ.
Note that End G is a multiplicative submonoid of GG . Clearly,
2.3.2. ∀α, β ∈ GG ∀γ ∈ End G
(α ∔ β)γ = αγ ∔ βγ.
Unless G is abelian, End G is not additively closed as idG ∔ idG ∈ End G if and only if
G is abelian. The following general criterion for a sum of two endomorphisms to be an
endomorphism of G is straightforward:
2.3.3. ∀α, β ∈ End G (α ∔ β ∈ End G ⇔ ∀x, y ∈ G y α xβ = xβ y α ).
For example, if U, T E G such that U ∩ T =  and α, β ∈ End G such that Gα ⊆ U,
Gβ ⊆ T , then α ∔ β ∈ End G. In the special case of 2.2, this holds for α = πU , β = ψT ,
and we have γ = πU ∔ ψT . An automorphism γ of a group G is called a replacement
automorphism (with respect to Ω) if there exist U, T, V E G such that the hypotheses
Ω
of the special case of 2.2 are satisfied and γ = πU ∔ ψT .22 By 2.2 we then also have
γ = idG ∔ ψT πU . We write RΩ G for the subgroup of Aut G generated by all replacement
automorphisms of G. In 2.2 (where U E G), the subgroup T πU is the image of an
Ω-homomorphism of G/U into Z(U)(≤ Z(G)) as G/U ∼
= T . We set
Ω
ZG,Ω :=
Y
{Gζ |ζ ∈ EndΩ G, Gζ ≤ Z(U), U ζ =  for some direct Ω-factor U of G}.
By 2.2, every replacement automorphism centralizes G/ZG,Ω. Hence we have
2.3.4. For any set of operators Ω on G, [G, RΩ G] ≤ ZG,Ω ≤ Z(G). If for all bipartite
direct Ω-decomposition (U, V ) of G there is no nontrivial Ω-homomorphism of V into
Z(U), then RΩ G = .
If n ∈ N and (α1 , . . . , αn ) is an n-tuple of endomorphisms of G such that Gαi , Gαj
commute elementwise for any distinct i, j ∈ n, we conclude inductively from 2.3.3 that
P
j∈n αj is an endomorphism of G and independent of the order of summation.
A set Ω of operators of G is called normal if all Ω-subgroups of G are normal in G. If Ω is
normal, an Ω-decomposition of G is a set ψ of Ω-invariant subgroups of G such that G is
22
The effect of γ is a “replacement” of the direct factor V (in the given Ω-decomposition of G) by T .
26
Q
their (restricted) direct product. If T is one of them and T ∗ := (ψ r{T }), then (T ∗ , T )
is a special case of a bipartite Ω-decomposition of G. The corresponding projection
onto T is an Ω-epimorphism (2.1.1). No confusion will arise if this epimorphism is
denoted by ψT . This is obviously an idempotent23 element of EndΩ G, and ker ψT = T ∗ .
24
ψT
The projections ψT where T ∈ ψ are pairwise
P orthogonal , and there images G
(T ∈ ψ) commute elementwise. If ψ is finite, T ∈ψ ψT = idG . More generally,
for any
P
element
V of a direct Ω-decomposition π of G we then have, by 2.3.2, T ∈ψ ψT |V πV =
P
( T ∈ψ ψT |V )πV = idV .
Let Ω be a normal set operators of G. An Ω-factor V of G ist called retracting if V 6=  and
for every idempotent ϕ ∈ EndΩ G such that Gϕ = V and for every finite Ω-decomposition
ψ of G there exists an element T ∈ ψ such that ψT |V ϕ is an automorphism of V . Clearly,
every retracting Ω-factor of G is Ω-indecomposable.
2.4 Lemma. Let G be a group with a normal set of operators Ω, ψ a finite Ω-decomposition of G into Ω-indecomposable subgroups.
(1) Let (U, V ) be an Ω-decomposition of G, V retracting. Then there exists γ ∈ RΩ G
such that V γ ∈ ψ, [U, γ] = .
(2) Let π be an Ω-decomposition of G, π0 a subset ofQ
π consisting of retracting Ω-factors.
Then there exists α ∈ RΩ G such that π0α ⊆ ψ, [ (π r π0 ), α] = .
Proof. (1) Put π := (U, V ). Let T ∈ ψ such that ψT |V ϕ ∈ Aut V. We claim that
πV |T is an isomorphism of T onto V.
(∗)
T
-V
πV
V
ψT
Ω
-N
K

-
Put N := V ψT , K := T ∩ker πV . Then V ∼
= V.
= N ≤ T , T /K ∼

Ω
Ω
Since ψT |V πV ∈ Aut V it follows that N ∩ K = , NK = T ,
˙ K. But T is Ω-indecomposable and N ∼
hence T = N ×
= V 6= .
Ω
Thus K = , N = T , proving (∗).

By 2.1.2, V , T are Ω-complements of U in G. By 2.2, there is a replacement automorphism γ such that V γ = T and [U, γ] = .
(2) For finite sets π0 we prove the claim by induction on |π0 |: If π0 = ∅, put α := idG .
Now
exists β ∈ RΩ G such that (π0 r {V })β ,
Q let V ∈ π0 and assume that
Q there
β
[V (π r π0 ), β] = . Put U := (π r {V }). Then (U, V ) is an Ω-decomposition
of G. Choose γ by means of (1) and set α := βγ. Then α ∈ RΩ G and
Y
Y
π0α = (π0 r{V })βγ ∪{V βγ } = (π0 r{V })β ∪{V γ } ⊆ ψ, [ (πrπ0 ), α] = [ (πrπ0 ), γ] = 
Hence the claim holds if π0 is finite. But ψ is finite so that, by what we have proved, π0
contains at most |ψ| elements.
23
24
i. e., it coincides with its square.
i. e., ψT ψT ′ = 0 for any two distinct T, T ′ ∈ ψ.
27
If we assume the hypotheses of 2.4 and write πr for the set of all retracting elements
of an Ω-decomposition π of G into Ω-indecomposable subgroups (analogously ψr for ψ),
then πrα = ψr for some α ∈ RΩ G. The case where πr = π is of major importance:
2.5 Theorem (General Krull-Schmidt theorem). Let G be a group with a normal set of
operators Ω. Suppose that G has an Ω-decomposition into retracting Ω-subgroups. Then
RΩ G acts transitively on the set of all finite Ω-decompositions of G into Ω-indecomposable
subgroups.25
In particular, under the hypotheses of 2.5 all members of a finite Ω-decomposition of G
into Ω-indecomposable subgroups are retracting. From 2.3.4 we conclude:
2.6 Corollary. Let G be a group with a normal set of operators Ω. Suppose that for
any Ω-factor U of G there is no nontrivial Ω-homomorphism of G/U into Z(U). (In
particular, this is the case if there is no nontrivial Ω-homomorphism of G into Z(G).)
Then there exists at most one finite Ω- decomposition of G into retracting subgroups. Note that the notion of a retracting Ω-factor V is, by definition, a relative one, depending
on (the finite Ω-decompositions of) the group G and not only on V as a group with
operator set Ω. By contrast, the property of being Ω-indecomposable is an absolute one,
depending only on the Ω-factor in question, not on G. The following remark, however,
shows that certain (internal) structural properties of an Ω-factor imply that it must be
retracting – regardless of other Ω-factors of G, in this sense: “absolutely” retracting.
2.6.1. Let G be a group with a normal set of operators Ω, V an Ω-factor 6=  of G. If
every element of EndΩ V is either nilpotent26 or an automorphism, then V is retracting.
Proof. Let ϕ ∈ EndΩ G beP
an idempotent such that Gϕ = V and ψ be a finite Ωdecomposition of G. Then T ∈ψ ψT |V ϕ = idV . Assume that no summand is an automorphism of V . Then each summand is nilpotent, by hypothesis. From 2.3.3 it follows
that every
subsum is an Ω-endomorphism of V . Let ψ0 be a minimal subset of ψ such
P
that T ∈ψ0 ψT |V ϕ is an automorphism α of V . Then |ψ0 | ≥ 2. Let U ∈ ψ0 und set
P
β := ψU |V ϕα−1 . The choice of ψ0 implies that β and idV − β = T ∈ψ0 r{U } ψT |V ϕα−1
are nilpotent Ω-endomorphisms of V . They obviously commute, hence
n X
n k
n
β (idV − β)n−k .
∀n ∈ N idV = (β ∔ idV − β) =
k
k=0
But for all k ∈ n ∪ {0} we have β k = o or (idV − β)n−k = o if n is large enough, a
contradiction.
We mention important special cases where an Ω-factor turns out to be retracting:
25
The group RΩ G may be replaced by RΩ̂ G where Ω̂ is the set of all endomorphisms of G which leave
all Ω-factors of G invariant. Note that In G ⊆ RΩ̂ G.
26
i. e., an element a certain power of which equals o.
28
(A) (Azumaya’s condition) Let K be a commutative unitary ring, A a K-space and W
an A-module over K (see p. 23). Let V be a non-zero direct A-summand of W such
that the ring EndA V is local27 . Then V is retracting.
(B) (Krull-Schmidt chain condition) Let G be a group with a normal set of operators Ω, V be an Ω-indecomposable Ω-factor of G satisfying the ascending and the
descending chain condition28 for Ω-subgroups. Then V is retracting.
Clearly, (A) is a special case of 2.6.1. Applying 2.5, we obtain:
2.7 Corollary (Krull-Schmidt-Azumaya theorem). Let K be a commutative
ring, A a K-space and W an A-module over K. Suppose that there exists
direct decomposition of W into A-submodules V with the property that EndA V
Then AutA W acts transitively on the set of all finite direct decompositions of
indecomposable A-submodules.
unitary
a finite
is local.
W into
To prove (B) it suffices to show
2.8 Proposition (Fitting’s Lemma). Let V be a group with a normal set of operators Ω,
ϕ ∈ EndΩ V . Suppose that V satisfies the ascending and the descending chain condition
j
for Ω-subgroups. Then there exists a positive integer j such that V = V ϕ × ker ϕj . If V
is Ω-indecomposable, then ϕ is either nilpotent or an automorphism.
2
Proof. The two chains of Ω-subgroups V ≥ V ϕ ≥ V ϕ ≥ · · · ,  ≤ ker ϕ ≤ ker ϕ2 ≤ · · ·
j
must terminate after a finite number of steps. Hence there exists j ∈ N0 such that V ϕ =
j+1
V ϕ = · · · , ker ϕj = ker ϕj+1 = · · · . Put τ := ϕj . Then V τ /(V τ ∩ker τ ) ∼
= (V τ )τ = V τ ,
τ
hence V ∩ ker τ =  because otherwise we would obtain a non-terminating ascending
chain of Ω-subgroups of V τ . Furthermore, (V τ )τ = V τ implies that for all v ∈ V there
2
exists w ∈ V such that v τ = w τ , hence (w −1 )τ v ∈ ker τ . It follows that V τ ker τ = V .
j
If V is Ω-indecomposable, this means that either ker ϕj = V , V ϕ =  or ker ϕj = ,
j
V = V ϕ , i. e., ker ϕ = , V = V ϕ . Hence ϕ is nilpotent or an automorphism.
We apply 2.5, observe that RΩ̃ G ≤ AutΩ G (with Ω as in footnote 25) and hence obtain
2.9 Corollary (Classical Krull-Schmidt theorem). Let G be a group with a normal set of
operators Ω. Suppose that G satisfies the ascending and the descending chain condition
for Ω-subgroups. Then AutΩ G acts transitively on the set of all Ω-decompositions of G
into Ω-indecomposable subgroups.
Note that the hypotheses of 2.9 imply also the existence of a finite Ω-decomposition into
Ω-indecomposable subgroups.
27
28
i. e., the set of non-units of EndA V is additively closed.
A partially ordered set X satisfies the ascending chain condition (descending chain condition resp.)
if every chain in X has a greatest (smallest resp.) element. By Zorn’s Lemma, the ascending
(descending resp.) chain condition holds in X if and only if every nonempty subset of X contains a
maximal (minimal resp.) element. – In our context, we consider the set X of all Ω-subgroups of V
which is partially ordered by set-theoretic inclusion.
29
If Ω = In G, an Ω-factor is just a direct factor of the group G. A finite abelian group
is the direct product of its nontrivial Sylow subgroups, hence can only be directly indecomposable if it is a p-group for some prime p. Therefore the following result shows
that the only directly indecomposable finite abelian groups are the cyclic p-groups.
2.10 Proposition. Let p be a prime and A a finite abelian p-group.
(1) If x ∈ A is an element of maximal order, then hxi is a direct factor of A.
(2) A is directly indecomposable if and only if A is cyclic.
Proof. (1) We proceed by induction on |A|, the case |A| ≤ p being trivial. Let |A| > p
and x ∈ A of maximal order, X := hxi, Y be the subgroup of order p of X. If X = A
there is nothing to prove. Therefore let X < B ≤ A such that |B/X| = p. We claim
B contains a subgroup 6= Y of order p.
(∗)
By the choice of x, the image of the homomorphism ϕ : B → X, g 7→ g p , does not
contain x, hence is a proper subgroup of X. In particular, |B/ ker ϕ| < |X| = |B/Y | so
that ker ϕ > Y , implying (∗).
B
X
p
Y
p

T
A
Thus there is a subgroup T 6= Y of B of order p. It follows that B/T =
hT xi is a cyclic subgroup of A/T of maximal order as o(T x) = o(x)
and o(T a) ≤ o(a) for all a ∈ A. Inductively there exists a subgroup
˙
˙
C ≥ T of A such that A/T = B/T ×C/T
. It follows that A = X ×C.
(2) If A is cyclic, the subgroups of A form a chain, hence A is directly indecomposable.
If A is directly indecomposable, then A is cyclic, by (1).
An endomorphism of a finite cyclic p-group A either maps A into its maximal subgroup
and hence is nilpotent or is an automorphism. By 2.6.1, it follows that a direct factor
which is a finite cyclic p-group is always strongly indecomposable. An arbitrary finite
group G trivially has a direct decomposition into directly indecomposable subgroups.
For an abelian p-group G this means that G ∼
= Cpk1 × · · · × Cpkn for some k1 , . . . , kn ∈ N
where Cj denotes a cyclic group of order j (j ∈ N0 ). Without loss of generality we may
assume that k1 ≥ · · · ≥ kn . As the groups Cpi are strongly indecomposable, we obtain
the following consequence of 2.5:
2.11 Theorem (Structure theorem for finite abelian p-groups). Let p be a prime, G a
finite abelian p-group. Then
G∼
= C p k1 × · · · × C p kn
for uniquely determined numbers n ∈ N0 , k1 , . . . , kn ∈ N such that k1 ≥ · · · ≥ kn .
The n-tuple (pk1 , . . . , pkn ) is called the type of the abelian p-group G.29 If |G| = pm , we
have 1 ≤ n ≤ m. G is cyclic if and only if n = 1. In the opposite extremal case, n = m,
29
If m ∈ N and k1 , . . . , kn are positive integers such that k1 ≥ · · · ≥ kn and k1 + · · · + kn = m, the
n-tuple (k1 , . . . , kn ) is called a partition of m. By 2.11, the number of abelian p-groups of order pm
equals the number of partitions of m.
30
every element 6= 1G is of order p, k1 = · · · = km = 1; G is then called elementary abelian.
Elementary abelian groups play an important role on many occasions in group theory.
One of the reasons for this is the following proposition.
2.12 Proposition. Let H be a finite30 minimal normal subgroup of a group. Then there
× · · · × S for some r ∈ N.
exists a simple group S such that H ∼
=S
|
{z
}
r
Proof. Let G be the given group, S a minimal normal subgroup of H. For all g ∈ G we
have S ∼
= S g E H, in particular
min
(∗)
∀N E H
S g ⊆ N or S g ∩ N = .
Q
Q
g
We have H ⊇ g∈G S g E G, hence g∈G SQ
= H. As H is finite there exists a minimal
(clearly finite) subset T of G Q
such that g∈T S g = H. For every x ∈ T we have
Q
g
x
g
g
g∈T r{x} S ⊳ H, hence S ∩
g∈T r{x} S = , by (∗). It follows that {S |g ∈ T } is
a direct decomposition of H. Since every normal subgroup of a direct factor of H is
normal in H, it follows that S is simple as S was chosen as a minimal normal subgroup
of H. Hence the claim where r = |T |.
By 2.5, r is unique and S is unique up to isomorphism. Now H is either elementary
abelian or a direct product of nonabelian simple groups. In the latter case, H has a
unique decomposition into isomorphic simple direct factors (∼
= S), by 2.6. Every inner
automorphism of G induces a permutation of the set of simple direct factors of H. This
action of G is transitive because the product over a G-orbit of simple direct factors of
H is a normal subgroup of G, hence equals H by the minimality of H. It follows that
there is just one G-orbit, containing all simple direct factors of H. Summing up, we
have shown:
2.12.1. Let H be a finite nonabelian minimal normal subgroup of a group G. Then there
exists a unique finite direct decomposition π of H into isomorphic simple groups. The
action of G on π,
π → π
,
G → Sπ , g 7→
S 7→ S g
T
is transitive. Its kernel is S∈π NG (S).
We now return to the general situation of an arbitrary group G with operator set Ω.
2.13 Definition. Let G be a group, U ≤ G. A subnormal series from U to G is a finite
set K of subgroups U0 , . . . , Un of G such that U = U0 ⊳ U1 ⊳ · · · ⊳ Un = G. U is called
subnormal in G, denoted by U EE G, if such a series exists.31 For every such chain K we
set U i := Ui /Ui−1 for all i ∈ n and K := {U i |i ∈ n}. Furthermore, l(K) := n = |K| is
called the length of K. If V E G for all V ∈ K, K is called a normal series in G.
30
31
The proof shows that the hypothesis of finiteness of H may be considerably weakened.
In contrast to normality, subnormality is a transitive relation on the set of subgroups of a group.
31
Let Ω be a set of operators on G. A subnormal Ω-series is a subnormal series K consisting
of Ω-subgroups. If U E V ≤ G, V /U is Ω-simple and there exists a subnormal Ω-series
Ω
Ω
from V to G, then V /U is called an Ω-composition factor of G. An Ω-composition series of
G is a subnormal Ω-series K from  to G such that every element of K is an Ω-composition
factor of G. As usual, the prefix “Ω-” is dropped if Ω = ∅. In the special case where
Ω = G acts by conjugation on G, an Ω-composition series of G is called a chief series of
G and an Ω-composition factor is then a chief factor of G.
2.13.1. If a group G satisfies the ascending and the descending chain condition for
subnormal Ω-subgroups, there exists an Ω-composition series of G.
Proof. We shall make use of the equivalences mentioned in footnote 28. Let X be the
set of all subnormal Ω-subgroups V of G such that there exists an Ω-composition series
of V . Then X 6= ∅ as  ∈ X. Let W be a maximal element of X. We claim that W = G:
Otherwise W < G so that the set of all subnormal Ω-subgroups properly containing W
is non-empty, hence has a minimal element W̄ . If K is an Ω-composition series of W ,
then K ∪ {W̄ } is an Ω-composition series of W̄ , a contradiction since W < W̄ .
2.13.2. Let K be a subnormal Ω-series of G, H ≤ G, N E G. Then
Ω
Ω
K∩H := {V ∩ H | V ∈ K} is an Ω-subnormal series of H,
K≡N := {V N/N | V ∈ K} is an Ω-subnormal series of G/N.
If K is an Ω-composition series of G, K≡N is an Ω-composition series of G/N and K∩H
is an Ω-composition series of H if H EE G. Moreover, l(K) = l(K∩N ) + l(K≡N ).
Proof. If U ⊳ V for some subnormal Ω-subgroup V of G, then
Ω
either U ∩ H = V ∩ H or U ∩ H ⊳ V ∩ H, UN = V N or UN ⊳ V N.
Ω
Ω
Suppose that V /U is Ω-simple. Then either V N/UN ∼
= V /U or
UN
G
N
Ω
V
V N = UN. If H is subnormal in G, then a subgroup of H is
subnormal in H if and only if it is subnormal in G. Therefore
either (V ∩ H)/(U ∩ H) ∼
= V /U or V ∩ H = U ∩ H. Moreover,
U
Ω
the Ω-simplicity of V /U implies that U(V ∩ N)(= UN ∩ V ) either
coincides with V or with U. We have V ∩ N = U ∩ N if and only
if V N 6= UN. The equation on the lengths follows.
VN

The assertions of 2.13.2 with respect to intersections become false if the term “Ωcomposition series” is replaced by “chief series” because the intersections of the members
of a chief series of G with N in general do not form a chief series of N: There may exist
normal subgroups of N between those intersections which are not normal in G. On the
other hand, from a chief series of G clearly a chief series of G/N is obtained by factorizing modulo N.
Subnormal Ω-series K, K∗ of G are called similar, denoted by K ∼ K∗ , if there exists a
32
bijection β of K onto K∗ such that U β ∼
= U for all U ∈ K r {}. Clearly, ∼ is an equiΩ
valence relation on the set of all subnormal Ω-series of G. Instead of comparing direct
decompositions we now compare subnormal series of a group and obtain the following
main result:
2.14 Theorem (Jordan-Hölder). Any two Ω-composition series of a group with operator
set Ω are similar.
Proof. We proceed by induction on the sum of the lengths of the two Ω-composition
series. If one of the two lengths – a fortiori if their sum – equals 0 , the considered
group is of order 1 and the claim trivial. For the inductive step, let G be the group
with the two Ω-composition series K, K∗ in question, l(K), l(K∗ ) > 0. Set K̇ := K r {G},
K̇∗ := K∗ r {G}, N := max K̇, N ∗ := max K̇∗ . Then N E G, N ∗ E G, the quotients G/N,
Ω
∗
Ω
∗
G/N are Ω-simple, and K̇, K̇ are Ω-composition series of N, N ∗ resp.
If N = N ∗ , then inductively K̇ ∼ K̇∗ , hence K = K̇ ∪ {G} ∼ K̇∗ ∪ {G} = K∗ .
If N 6= N ∗ , put D := N ∩ N ∗ . Then NN ∗ = G, G/N ∼
= N/D.
= N ∗ /D, G/N ∗ ∼
Ω
N∗
N
Ω
By 2.13.2, K̇∩D , K̇∗∩D are Ω-composition series of D; K̇, (K̇∗∩D ) ∪ {N} are
Ω-composition series of N the lengths of which have a sum < l(K) + l(K∗ ),
likewise K̇∗ , (K̇∩D ) ∪ {N ∗ } with respect to N ∗ . By our inductive hypothesis,
K̇∩D ∼ K̇∗∩D , K̇ ∼ (K̇∗∩D ) ∪ {N}, K̇∗ ∼ (K̇∩D ) ∪ {N ∗ }. It follows that
G
D

K = K̇ ∪ {G} ∼ K̇∗∩D ∪ {N, G} ∼ K̇∩D ∪ {N ∗ , G} ∼ K̇∗ ∪ {G} = K∗ .
In particular, all chief series of a group have the same length and consist, up to Gisomorphisms, of the same chief factors. Furthermore, the same stement holds for all
composition series of a group and their composition factors. It should be noted, however,
that both the general Krull-Schmidt theorem 2.5 and the Jordan-Hölder theorem 2.14
are uniqueness statements, not existence statements regarding decompositions, composition series resp.
If we write P(K) for the direct product of all groups in K for a subnormal series K of G,
the Jordan-Hölder theorem may be re-formulated as follows:
2.14’ P(K) ∼
= P(K∗ ) for any Ω-composition series K, K′ of G.
Ω
Clearly, 2.14 implies 2.14’. Conversely, 2.14’ implies 2.14 by 2.5 as K, K∗ are (up to
isomorphism) Ω-decompositions of P(K) into Ω-simple groups.
Finally we observe that the choice of  as the subgroup where the series begins is not
necessary in 2.13.1–2.14: With trivial modifications of the proof we obtain the assertion of the Jordan-Hölder theorem more generally with respect to any two not refinable
subnormal Ω-series from a given subnormal Ω-subgroup U of G to G.
33
3 Complements
Let G be a group and (U, V ) a semidirect decomposition of G. Then U acts on V
by conjugation, and the multiplication in V , the multiplication in U and this action
′
determine the operation of the group G as we have uvu′v ′ = uu′v u v ′ for all u, u′ ∈ U,
v, v ′ ∈ V . Conversely, this observation gives rise to an important concept of a group
construction:
3.1 Definition. Let U, V be groups and f a group action of U on V . We define an
operation · on the cartesian product U × V of the sets U, V by
f
∀u, u′ ∈ U ∀v, v ′ ∈ V
′
(u, v) · (u′ , v ′ ) := (uu′ , v u f v ′ ).
f
′
′
′
Obviously, (1U , 1V ) is neutral. The equation (v uf v ′ )u f = v (uu )f v ′u f shows that · is
f
−1
−1
−1 u f
associative, and (u , (v )
) is an inverse of (u, v). Hence U × V is a group with
respect to · called the semidirect product of V with U with respect to the action f
f
and denoted by U ⋉ V or V ⋊ U. The natural injections ε : u 7→ (u, 1V ) (u ∈ U),
f
f
ι : v 7→ (1U , v) (v ∈ V ), are monomorphisms (observing that 1U f = idV ). We have
∀u ∈ U ∀v ∈ V
(1U , v) · (u, 1V ) = (u, v uf ) = (u, 1V ) · (1U , v uf ),
f
f
implying that (U ε , V ι ) is a semidirect decomposition of (U × V, · ), and showing that
f
ι uε
(v )
uf ι
= (v ) for all v ∈ V , u ∈ U:
3.1.1. The conjugation of V ι by elements of U ε is given by the action f .
If there is no doubt about f , just the symbols ⋉, ⋊ resp., are used32 . The direct
product of U and V occurs in the case of the trivial action f : U → {idV }. The
concept of semidirect product is of fundamental importance in group theory. Frequently
semidirect products are used in proofs and are constituents of more complicated group
constructions. Therefore it is awkward to carry along throughout the variables ε, ι.
For notational simplicity, it is common use to consider the groups U, V as they are
as subgroups of their semidirect product, i. e., instead of uε , v ι one simply writes u, v
resp. A necessary condition to do this is, of course, that the sets U, V have one and
only one element in common: their neutral one. It is usually assumed without further
comment that this condition is satisfied (or that the sets U, V may be arranged this way
32
˙ ι.
involving the “normal subgroup triangle” in the way that it correctly refers to V . – U ⋉ V = U ε ⋉V
34
without problems) when semidirect products occur. Then (U, V ) (instead of (U ε , V ι ))
is a semidirect decomposition of U ⋉ V .33 A group G is called a split extension of V
f
by U if there exists a normal subgroup N and a complement H of N in G such that
N∼
= V, H ∼
= U. We know that the isomorphism type of a split extension is completely
determined by the isomorphism type of the normal subgroup, its complement and the
action of the latter on the former by conjugation. Therefore, theorems are important
which (under appropriate hypotheses) allow the conclusion that a certain given group
is a split extension: With this information frequently a complicated group structure is
reduced to less complex (“smaller”) constituents of the group, thus its analysis reduced
to less complicated cases. Before we prove certain for this reason important “splitting
theorems” we give a number of examples of semidirect products.
Examples.
(1) Let n ∈ N>1 . Every subgroup generated by a transposition i j is a complement
of the normal subgroup An of Sn . Hence Sn ∼
= C2 ⋉ An . For n ≥ 5 it is an easy
consequence of 1.3 that , An , Sn are the only normal subgroups of Sn . Therefore,
apart from choosing different complements of An , there are no other non-trivial
possibilities to write Sn as a semidirect
product. It should
be noted, however,
that for n ≥ 6, the subgroups h 1 2 i and h 1 2 3 4 5 6 i, for example, are
complements of An which are not conjugate under Sn (see 1.1.4(1)). In this sense,
there exist “substantially different” complements of An in Sn .
The group S4 has not only the nontrivial normal subgroup A4 but also an elementary
abelian normal subgroup of order 4 which has 4 (conjugate) complements in S4 , the
four point stabilizers which are isomorphic to S3 (cf. 1.2(3) and the comment after
the proof of 1.2). Thus we also have S4 ∼
= S3 ⋉ (C2 × C2 ). Correspondingly, A4 ∼
=
C3 ⋉ (C2 × C2 ). (Here we did not bother to write down the respective group actions
f , but we know that these are given by conjugation within Sn after embedding the
factors of the abstract semidirect product.)
(2) For every abelian group V the mapping α : V → V , v 7→ v −1 , is an automorphism,
and is of order 2 if and only if V is not an elementary abelian 2-group. Let f
be the homomorphism of C2 into Aut V such that C2 f = hαi. The corresponding
semidirect product C2 ⋉ V is called the dihedral group of V . Usually the term refers
f
33
Frequently this process of sparing the variables ε, ι is called “identification” of U, U ε , of V, V ι resp.
This at the first sight rather mysterious word obtains an exact meaning as follows: If the groups given
are arranged in a way that the sets U, V satisfy the necessary condition that U ∩ V = {1U } = {1V },
then ε ∪ ι is an injection of the set U ∪ V into the set U × V and a monomorphism of the group U
and of the group V into the group U ⋉ V . Then (by an application of the extension principle, p. 4)
f
there exists a set W , an operation on W and an extension of ε ∪ ι to an isomorphism of W onto
U ⋉ V . Clearly, instead of the latter we may now consider the group W with its subgroups U , V .
f
This describes the normally tacitly assumed passage to a semidirect product which contains (not
only up to isomorphism) the given groups U , V as desired.
35
only to the case where V is cyclic, in particular, if only the order is given and no
isomorphism type is mentioned. Thus the notation D2k refers to the dihedral group
of order 2k which arises as the semidirect product C2 ⋉ Ck . It is of order 2 if k = 1,
f
elementary abelian (of order 4) if k = 2, isomorphic to S3 if k = 3, one of the two
types of nonabelian group of order 8 if n = 4.34 It is an easy exercise to prove that
a group of order 2p (p a prime) must be isomorphic either to C2p or to D2p .
(3) If V is a group and U ≤ Aut V , then U acts “by nature” on V , i. e., we may choose
f = idU for our group action and obtain a semidirect product of V with U. We
know that there are canonical embeddings of V , U into V ⋉ U, and conjugation of
an element of [the embedded] V by an [embedded] element α of U is simply obtained
by applying the automorphism α. The special semidirect product V ⋉Aut V is called
the holomorph of V .
(4) Let f be an action of a group U on a set X, M any group. We are going to show that
f induces canonically an action of U on the group (M X , ∔) (see 2.3), thus giving rise
←
−
to a semidirect product of M X with U. Defining ϕ u to be the composition (u−1 f )ϕ
− as a mapping of M X into M X , and we have
for all u ∈ U, ϕ ∈ M X , we read ←
u
∀u, v ∈ U ∀ϕ ∈ M X
(∗)
←
− ←
−
←
−
(ϕ u ) v = (v −1 f )(u−1 f )ϕ = ϕuv .
In the following, we shall write xu instead of x(uf ) for any x ∈ X, u ∈ U. For
←
−
←
−
←
−
all ϕ, ψ ∈ M X , x ∈ X we have x(ϕ ∔ ψ) u = (xu−1 ϕ)(xu−1 ψ) = x(ϕ u ∔ ψ u ). If
←
−
ϕ u = ȯ, i. e., xu−1 ϕ = 1M for all x ∈ X, it follows that ϕ = ȯ as Xu−1 = X.
←−1
−−
←
−
− is an
Given ψ ∈ M X , put ϕ := ψ u . Now (∗) first implies that ϕ u = ψ whence ←
u
automorphism of the group M X . Secondly it shows that
−,
U → Aut M X , u 7→ ←
u
is a homomorphism. The semidirect product U ⋉ M X based on this action of U on
M X is called the wreath product of U and M with respect to f . Its common notation,
M ≀ U, alternatively U ≀ M, takes into account the fact that it is canonically given
f
f
by the homomorphism f of U into SX from the very beginning. Explicitly, its
operation reads as follows: 35
∀u, v ∈ U ∀ϕ, ψ ∈ M X
(u, ϕ) · (v, ψ) = (uv, (v −1f )ϕ ∔ ψ)
f
Most natural choices for f are the actions ρ, λ of U on itself (X = U) by right
multiplication, inverted left multiplication resp.. The wreath products of U and M
34
The other one, the so-called quaternion group of order 8 (the multiplicative closure of the 4 standard
basis vectors of the quaternion algebra over R), has a unique subgroup of order 2, hence has no
nontrivial semidirect decomposition. – D8 has 5 subgroups of order 2. Four of them are non-normal
and each of these is a complement of two non-isomorphic normal subgroups of order 4.
35
A more general form of the wreath product arises when a group action ˜ of U on M is given and the
product ˜· is defined by (u, ϕ)˜· (v, ψ) = (uv, (v −1 f )ϕ ∔ ψũ−1 ) for all u, v ∈ U, ϕ, ψ ∈ M X .
f
f
36
for these choices are easily seen to be isomorphic and called regular. The regular
wreath product of U and M is denoted by M ≀ U or U ≀ M. In the sequel we will
prefer the symbol ≀ .
We take a look at the special case where X is finite, w. l. o. g. X = n where
n ∈ N. Then M X = M n , the set of all n-tuples over M.36 The elements of
U ≀ M have the form (u, (m1 , . . . , mn )) (where u ∈ U, mj ∈ M) which is the product
f
(u, (1M , . . . , 1M )) (1U , (m1 , . . . , mn )). No confusion will arise if we write this product
simply as u(m1 , . . . , mn ) – just dropping the 1’s which formally serve to make the
distinction between U, M n resp., and their canonical embeddings in U ≀ M. We
f
know that conjugation of an element of M n by an element of u ∈ U is given by
−, i. e.,
executing ←
u
∀m1 , . . . , mn ∈ M ∀u ∈ U
←
−
(m1 , . . . , mn )u = (m1 , . . . , mn ) u = (m1u−1 , . . . , mnu−1 ).
Thus the i-th component of an n-tuple over M is sent to the (iu)-th position. The
complete wreath product arises when U = Sn , f = id. We extend M by one “new”
element 0 (∈
/ M), setting M0 := M ∪ {0}, and define
Sn ≀ M
Φ:
id
→ M0n×n
π(m1 , . . . , mn ) 7→
(
mj
where aij =
0
(aij )
if j = iπ
.
otherwise
The matrix associated with π(m1 , . . . , mn ) is graphically obtained as follows:

i = jπ −1








→





π(m1 , . . .
...,
↓
0
mj , . . .
...,
...
0
0
..
.
..
.
0
mj
0
..
.
0
0
mn )
↓
↓
...
0















For each j, the element
mj is pulled down to
the i-th place in the jth column, where i =
jπ −1 . All other matrix
entries are 0.
Thus the index of the row containing the element mj is determined by π. Conversely these indices clearly determine π, while the n-tuple (m1 , . . . , mn ) is trivially
read from the matrix (by “ignoring all zeros”). In particular, Φ is injective. Every
matrix in the image of Φ has exactly one entry different from 0 in every row and in
36
Recall that, by definition, an n-tuple over M is a function of n into M . The function which maps j
to mj ∈ M for all j ∈ n is denoted by (m1 , . . . , mn ).
37
every column. A matrix with this property is called monomial.37 We now make M0
into a monoid with zero by extending the operation of M as follows: a · 0 := 0 =: 0 · a
for all a ∈ M0 . Moreover, we set a + 0 := a =: 0 + a for all a ∈ M0 . This allows
us to multiply monomial matrices over M0 via the usual matrix multiplication rule.
We show that Φ is a homomorphism: Let m1 , . . . , mn , r1 , . . . , rn ∈ M, π, σ ∈ Sn
and aij , bij ∈ M0 such that (π(m1 , . . . , mn ))Φ = (aij ), (σ(r1 , . . . , rn ))Φ = (bij ). If
aik bkj 6= 0 for some i, j, k ∈ n, it follows that k = iπ, j = iπσ and aik = miπ ,
bkj = riπσ . Given i, j, this happens for at most one k. Hence
(
X
mjσ−1 rj if iπσ = j,
∀i, j ∈ n
aik bkj =
0
otherwise.
k∈n
As π(m1 , . . . , mn ) σ(r1 , . . . , rn ) = πσ(m1σ−1 r1 , . . . , mnσ−1 rn ), this means that Φ is
a homomorphism. Summarizing, the complete wreath product Sn ≀ M is, up to
id
isomorphism, the group of all monomial n × n matrices over M0 . The image of
id(m1 , . . . , mn ) under Φ is the diagonal matrix diag[m1 , . . . , mn ] while the image of
π(1M , . . . , 1M ) is the permutation matrix (see footnote 37) associated with π.
We now prove that the automorphism group of the direct product of finitely many copies
of a simple non-abelian group may be described as a complete wreath product:
3.2 Proposition. Let S be a group, |S| =
6 1, n ∈ N, G := S × · · · × S. The mappings
n
G
→
G
∆ : (Aut S) × · · · × (Aut S) → Aut G, (α1 , . . . , αn ) 7→
,
n
(s1 , . . . , sn ) 7→ (sα1 1 , . . . , sαnn )
G
→
G
,
Π : Sn → Aut G,
π 7→
(s1 , . . . , sn ) 7→ (s1π−1 , . . . , snπ−1 )
are monomorphisms. Let V, S˜n denote the images of ∆, Π resp. Then
˙ ≤ Aut G
(1) Sn ≀ (Aut S) ∼
= S˜n ⋉V
id
˙ = Aut G.
(2) If S is simple and non-abelian, then S̃n ⋉V
Proof. The first claim (on ∆) is obvious. For all π, σ ∈ Sn we have πΠ, σΠ ∈ Aut G, and
σΠ
(s1 , . . . , sn )(πσ)Π = (s1σ−1 π−1 , . . . , snσ−1 π−1 ) = (s1π−1 , . . . , snπ−1 )σΠ = (s1 , . . . , sn )πΠ
for all s1 , . . . , sn ∈ S which proves that Π is a homomorphism. If π 6= id, there exists
an i ∈ n such that i < iπ. Then for any s ∈ S r  we have (. . . , 1S , . . . , s , . . . )πΠ =
(. . . . . . , 1S , . . . ), hence πΠ 6= idG . Thus Π is a monomorphism.
i
iπ
iπ
(1) Let αj ∈ Aut S, π ∈ Sn . If we assume that (s1π−1 , . . . , snπ−1 ) = (sα1 1 , . . . , sαnn ), i. e.,
37
A monomial matrix in which every non-zero entry equals 1 is called a permutation matrix. If M = {1},
Φ reduces essentially to an isomorphism of Sn onto the group of all permutation matrices which also
explains their name.
38
α
sjπ−1 = sj j for all s1 , . . . , sn ∈ S, we have π = id and αj = idS for all j ∈ n. It follows
that S̃n ∩ V = . Furthermore, the composition (π −1 Π)((α1 , . . . , αn )∆)(πΠ) maps every
α
α
n-tuple (s1 , . . . , sn ) over S to (sα1π1 , . . . , sαnπn )πΠ = (s1 1π−1 , . . . , snnπ−1 ), which means, in
other words, that
((α1 , . . . , αn )∆)πΠ = (α1π−1 , . . . , αnπ−1 )∆.
This equation shows that V is normalized by S̃n and, more precisely, that the bijection
˙
Sn ≀ (Aut S) → S̃n ⋉V,
π(α1 , . . . , αn ) 7→ πΠ((α1 , . . . , αn )∆)
id
is a homomorphism, hence an isomorphism.
(2) Let S be simple and non-abelian. By 2.6, G has a unique direct decomposition into
n normal subgroups S1 , . . . , Sn ∼
= S. Hence every automorphism γ ∈ Aut G induces a
permutation πγ of n, defined by the condition that Sjπγ = Sjγ for all j ∈ n. The mapping
F : Aut G → Sn , γ 7→ πγ ,
is a homomorphism. Now γ ∈ ker F if and only if Sjγ = Sj for all j ∈ n, i. e., if and
only if γ ∈ V . Hence Sn ∼
= (Aut G)F ≤ Sn which implies the
= S̃n V /V ≤ (Aut G)/V ∼
claim.
Now let G be an arbitrary group and π = (U, V ) a semidirect decomposition of G. We
know that the projection πU is a homomorphism of G onto U, but for the other projection
πV a different rule holds. The composition of πU with the action of the subgroup U on
the normal subgroup V given by by conjugation defines a group action f of G on V .
′
For all u, u′ ∈ U, v, v ′ ∈ V we have uvu′ v ′ = uu′ v u v ′ so that we obtain the following
property of πV :
∀g, h ∈ G (gh)πV = (gπV )hf (hπV ).
Thus πV may be viewed as a “disturbed homomorphism” because the calculation of the
value of a product depends on the action of the second factor.
3.3 Definition. Let G be a group which acts on a group M. Write h̃ for the automorphism of M associated with h ∈ G. A crossed homomorphism of G into M is a mapping
w : G → M such that
∀g, h ∈ G (gh)w = (gw)h̃(hw).
Clearly, such a mapping w is a homomorphism if and only if the action of G is trivial
on the image of w. Thus crossed homomorphisms are generalized homomorphisms. It
is therefore natural that certain properties of homomorphisms hold in a weakened form
for a crossed homomorphism w. Let ker w := {g | g ∈ G, gw = 1M }. A first trivial
consequence of the definition is that 1G ∈ ker w. We observe further useful properties.
3.3.1. ∀g ∈ G (gw)−1 = (g −1 w)g̃ ,
as 1M = (g −1g)w = (g −1 w)g̃ (gw) for all g ∈ G.
39
3.3.2. ker w ≤ G.
Proof. We know that 1G ∈ ker w. If g, h ∈ ker w, obviously gh ∈ ker w, and, by 3.3.1,
g −1 ∈ ker w.
3.3.3. ∀g, h ∈ G
hg −1 ∈ ker w ⇔ gw = hw.
g
−1
Proof. By 3.3.1, (hg −1 )w = (hw)g (g −1 w) = (hw)(gw)−1
(hw)(gw)−1 = 1M ⇔ gw = hw.
gg
−1
. Hence hg −1 ∈ ker w ⇔
3.3.4. Gw ≤ M ⇔ Gw is G-invariant.
Proof. We have 1M = 1G w ∈ Gw. Hence Gw ≤ M ⇔ ∀g, h ∈ G (gw)(hw)−1 ∈ Gw ⇔
∀g, h ∈ G (gh)w (hw)−1 ∈ Gw. As (gh)w (hw)−1 = (gw)h̃, the claim follows.
3.3.5. If M ∗ E M, then w ∗ : G → M/M ∗ , g 7→ M ∗ (gw), is a crossed homomorphism,
G
as (gh)w ∗ = M ∗ (gw)h̃ (hw) = (M ∗ (gw))h̃M ∗ (hw) = (gw ∗)h̃ (hw ∗ ) for all g, h ∈ G.
We now consider the special case where M = G and G acts by conjugation. The
elementary commutator equation [gh, x] = [g, x]h [h, x] (for any g, h, x ∈ G) then provides
a first and most natural type of example: For every x ∈ G, the mapping
G → G, g 7→ [g, x],
is a crossed homomorphism. A major difference to the general situation is that, given a
crossed homomorphism w of G into G, we may apply w again to the elements of Gw.
3.4 Proposition. Let w be a crossed homomorphism of G into G, K := ker w, B := Gw.
(1) Bw = B if and only if KB = G,
(2) w|B is injective if and only if Kb 6= Kb′ for any two distinct b, b′ ∈ B,
(3) w|B is a permutation of B if and only if B is a right transversal of K in G,
(4) If B E G and w|B is a permutation of B, then K is a complement of B in G.
Proof. (1) Let g ∈ G. If Bw = B, there exists an element b ∈ B such that gw = bw,
hence gb−1 ∈ K by 3.3.3. Thus g ∈ KB. Conversely, let x ∈ K, b ∈ B such that g = xb.
Then gw = (xw)b (bw) = bw ∈ Bw. – As for (2), it suffices to observe that, by 3.3.3,
Kg = Kh for any g, h ∈ G if and only if gw = hw. Combining (1) and (2), we obtain
(3). Finally, (4) is just one of the implications in (3) for the case where B E G.
Given a normal subgroup M of a group G, we will pursue the idea of finding a complement
of M in G in the form of the kernel of some crossed homomorphism of G into M.
Under appropriate hypotheses we will see that our crossed homomorphism induces a
permutation on M so that our aim will be reached thanks to 3.4(4). Finding a suitable
crossed homomorphism, however, wants a special idea and some further preparation. It
40
will, in its general form, provide a powerful group-theoretic instrument to obtain short
proofs for quite a number of important results, as will be seen in the sequel.
Let G, M be groups acting on a set X via group actions fG , fM resp., such that
GfG ≤ NSX (MfM ). By 1.5.4, we know that fG induces an action on X/M. A mere
re-formulation of this osservation arises when we consider sets of representatives for the
M-orbits in X instead of the M-orbits themselves:
1.5.4’ Let G, M be groups acting on a set X such that the action of M is normalized by
the action of G. Then the latter induces an action on the set of all sets of representatives
of X/M.
If we additionally assume that fM is injective, we obtain a group action of G on M by
defining
−1
.
∀m ∈ M ∀h ∈ G
mh̃ := (mfM )hfG fM
Now we assume that the hypotheses of 1.5.4’ are satisfied and that fM is elementwise
fixed-point-free. As long as there is no risk of confusion, we simply write xg instead of
x(gfG ), xm instead of x(mfM ). If B ∈ X/M, x, y ∈ B, there exists a unique m ∈ M
x
such that ym = x. This element of M will be denoted by .
y
3.4.1. If x, y, z ∈ B, h ∈ G, then
−1
x
y
=
x
y
zx
x
=
yz
y
h̃
xh
x
=
yh
y
(1)
(2)
(3)
Here the first two parts are obvious by the definition, and if m = xy , we have (yh)mh̃ =
yhh−1 mh = ymh = xh, proving (3).
3.5 Proposition. Let G, M be groups acting on a set X such that the action of M is
elementwise fixed-point-free and normalized by the action of G, Y := X/M. Let R be a
set of representatives of Y . For every B ∈ Y let rB the unique element of R ∩ B. Then
h̃
rBgh
rBgh
rBg
(1) ∀g, h ∈ G
=
,
rB gh
rB g
rBg h
(2) ∀g, h ∈ G
ϕgh = h−1 ϕg ∔ ϕh g̃ −1
where we define ϕg : Y → M, B 7→
38
38
rB g −1
, for all g ∈ G.
rBg−1
This should be read as an equation in the group (M Y , ∔) (see 2.3). If the actions of G and M even
centralize each other (i. e., g̃ = idM for all g ∈ G), the mapping g 7→ ϕg is a crossed homomorphism
of G into (M Y , ∔) with respect to the action of G on M Y as given (for U ) in 3.1, Ex. (4).
41
rBg h rBgh
, by 3.4.1(2), 3.4.1(3) resp.. (2) follows from
rB gh rBg h
rB h−1 g −1
−1
=
= (Bh−1 )ϕg ·(Bϕh )g̃ for all g, h ∈ G, B ∈ Y .
rBh−1 g−1
Proof. In (1), both sides equal
(1) and 3.4.1(1) as Bϕgh
3.6 Theorem. Let G be a group, M ≤ G, U := G/MG , g := MG g for all g ∈ G. Then
ω:
G → U ≀ M,
ρ̄
g 7→ gϕg ,
is a monomorphism (where ρ̄ is the action of U on G /. M by right multiplication, 1.4.5).
U
G

 R


M
Proof. We apply 3.5(2) to the case where X = G and G acts by right
multiplication ρ, M by inverted left multiplication. Then Y = G /. M and
⊂
g̃ = idM for all g ∈ G. For all h ∈ G, B ∈ Y we have B(h̄ρ̄) = Bh = B(hρ).
Hence, by the definition of the wreath product and by 3.5(2),
MG
←
−
∀g, h ∈ G gϕg hϕh = gh(ϕgh ∔ ϕh ) = ghϕgh .

Thus ω is a homomorphism. If g ∈ ker ω, then g ∈ MG and ϕg = ȯ, hence rB = rBg = rB g
for any B ∈ G ./ M, implying g = 1G .
If M E G, then M = MG so that ω in 3.6 is then a monomorphism into the regular
wreath product of G/M with M. We conclude:
3.7 Corollary. Let G be a group, M E G. Then G is isomorphic to a subgroup of
(G/M) ≀ M.
Let M EG, U := G/M. The (regular) wreath product U ≀ M is the semidirect product of
M U – a group which may be visualized as an “inflated” M, a direct product of “many”
M’s –, and U. It is certainly a remarkable property of the regular wreath product that
this particular split extension contains, up to isomorphism, any group with a normal
subgroup ∼
= U. For example, Cp ≀ Cp contains a cyclic subgroup of
= M and quotient ∼
order p2 , for any prime p. Inductively, it is easily seen that every group of order pn
is contained in the iterated wreath product ((Cp ≀ Cp ) ≀ Cp ) ≀ · · · (n “wreath product
factors”). The quaternion group Q8 is contained in C2 ≀ C4 and in C2 ≀ (C2 × C2 ).
Now assume the hypotheses of 3.5 and let X/M be finite. We write x ∼ y if the elements
M
x, y ∈ X represent the same orbit of M in X. For any sets R, S of representatives for
X/M set
Y
S
s
:=
M ′ ∈ M/M ′ .
R
r
(r,s)∈R×S
r∼s
M
If M is abelian,
S
R
may be considered as an element of M. Applying 3.4.1, we obtain
42
3.7.1. If R, S, T are sets of representatives for X/M, h ∈ G, then
(I)
(II)
(III)
−1
S
R
=
R
S
R
T R
=
S T
S
h̃
Rh
R
=
Sh
S
3.8 Proposition. Let G, M be groups acting on a set X such that the action of M is
elementwise fixed-point-free and normalized by the action of G. Suppose X/M is finite.
For any set of representatives R for X/M put
wR : G → M/M ′ , g 7→
R
.
Rg
(1) wR is a crossed homomorphism of G into M/M ′ ,
R
(2) hwS = hwR [ , h̃] for every h ∈ G and set S of representatives for X/M.
S
Proof. By 3.7.1, we have for all g, h ∈ G
h̃
R
R
Rh R
R
(gh)wR =
=
=
= (gwR)h̃ (hwR ),
Rgh
Rgh Rh
Rg
Rh
h̃
−1
Rh R S
R
R
R
S
=
=
hwR
= hwR [ , h̃].
hwS =
Sh
Sh Rh R
S
S
S
First application of 3.8. Let G be a group, M ≤ H ≤ G, M EG,
M abelian with a complement K in H, X := G /. K, n := |G : H|
finite. Both G and M act via right multiplication on X. The
action of M is elementwise fixed-point-free and normalized by the
action of G. The number of M-orbits in X equals n. Therefore, for
any m ∈ M the definition of wR in this case implies immediately
3.8.1. mwR = m−n for any set R of representatives for X/M.
G
H
n
M
ab.
K

Let g ∈ G. A coset A = Ka (a ∈ G) has the property that A and Ag represent the same
M-orbit if and only if Ag = Am for some m ∈ M, i. e., g ∈ K a M (= H a ). If H E G it
follows that, with respect to the action of G on X/M, exactly the elements of H have
all orbits of length 1.
3.8.2. If H E G and Ω is a set of G-endomorphisms of H which normalize K and M,
then wR |H is a crossed Ω-homomorphism of H into M.
43
Proof. ∀h ∈ H, ∀α ∈ Ω hα wR =
Q
A
A∈R Ahα
=
Q
A∈R
A α
Ah
= (hwR )α .
3.9 Theorem (Gaschütz 1952). Let G be a group, M an abelian normal subgroup of
G, M ≤ H ≤ G and n := |G : H| finite. Suppose the mapping M → M, m 7→ mn , is
bijective.39
(1) If H splits over M, then G splits over M.
(2) If L, L̃ are complements of M in G and L ∩ H, L̃ ∩ H are conjugate under M, then
L, L̃ are conjugate under M.
Proof. Choose X as above, with respect to an arbitrary complement of M in H. From
3.8.1 and Proposition 3.4(4) we obtain by our hypothesis on n:
(∗)
ker wR is a complement of M in G,
for any set R of representatives for the M-orbits in X. This proves (1).
M
L̃
K

sis,
R
S
Now assume the hypotheses of (2). Then K := L ∩ H is a complement
of M in H. Choosing R := L /. K as a set of representatives for X/M,
we see that L ⊆ ker wR as Rg = R for all g ∈ L, hence L = ker wR by
L
(∗). Clearly we may assume that L̃ ∩ H = K. Then the set S := L̃ /. K
again is a set of representatives for X/M, and L̃ = ker wS . By hypothefor some m ∈ M. Hence, by 3.8(2),
G
H
= m−n
R
∀g ∈ G gwS = gwR [ , g] = gwR (mwR )−1 (mwR )g
S
= (mg)wR (m−1 wR ) = (mgm−1 )wR ,
as M is abelian. Therefore, L̃ = ker wS = (ker wR )m = Lm .
3.10 Corollary (Maschke 1897). Let Ω be a commutative unitary ring, G a finite group,
H an ΩG-module, M an ΩG-submodule of H having an Ω-complement in H. If the
mapping M → M, m 7→ |G|m is bijective, M has an ΩG-complement in H.
Proof. Let G∗ = G ⋉ H, K an Ω-complement of M in H, R a set of representatives
for the M-orbits in G∗ /. K. Then |R| = |G∗ : H| = |G|. As H splits over M, ker wR
is a complement of M in G∗ , by (∗) in the proof of Theorem 3.9. By 3.8.2, wR|H is an
Ω-homomorphism of the abelian group H into M. Now H ∩ ker wR is an Ω-subspace
and normal in G∗ , hence an ΩG-complement of M in H.
The case where Ω is a field and char Ω ∤ |G| is known as Maschke’s theorem and a
ground-laying result in representation theory of finite groups. There exist technically
simpler proofs than the one given here, but still it is of some interest to have it as a side
result in our context.
If the group G in 3.9 is finite, then the mapping M → M, m 7→ mn , is bijective if and
39
A group M with this property is called uniquely n-divisible.
44
only if gcd(|M|, n) = 1. A subgroup H with the property that gcd(|H|, |G : H|) = 1
is called a Hall subgroup of G,40 and every abelian subgroup of it which is normal in G
satisfies the hypotheses in 3.9. In particular, Sylow subgroups are Hall subgroups, so
that we have the following typical application of 3.9:
3.11 Corollary. Let G be a finite group, H ∈ Syl(G), M an abelian normal subgroup
such that M ≤ H. If M has a complement in H, then M has a complement in G.
Complements of M in G are conjugate if and only if their intersections with H are
conjugate (under M).
Furthermore, the hypothesis in 3.9(1) is trivially satisfied if H = M. For finite groups,
this is the basic case of the following more general result:
3.12 Corollary. Let M be a soluble normal Hall subgroup of a finite group G. Then
(1) M has a complement in G,
(2) any two complements of M in G are conjugate under M.
′
for all j ∈ N0 .
Proof. Inductively we put, for any group H, H(0) := H, H(j+1) := H(j)
The solubility of M means that there exists a smallest index k ∈ N0 such that M(k) = .41
We show by induction on k that (1), (2) hold if M(k) = . This is trivial for k = 0. Let
k > 0 for the inductive step. M ′ is characteristic in M (cf. p. 23), hence normal in G.
By 3.9, there exists a unique conjugacy class of complements of M/M ′ in G/M ′ . Given
such a complement G1 /M ′ (where M ′ ≤ G1 ≤ G), M ′ is a soluble normal Hall subgroup
′
= M(k) = . Inductively, there exists a unique conjugacy class of
of G1 and M(k−1)
complements of M ′ in G1 . Any complement L1 of M ′ in G1 is a complement of M in
G, which completes the inductive step for the existence assertion. As for conjugacy, let
L2 be any complement of M in G. Put G2 := L2 M ′ . Then G2 /M ′ is a complement of
m
M/M ′ in G/M ′ , hence Gm
2 = G1 for some m ∈ M. It follows that L2 is a complement
′
of M ′ in G1 so that Lmm
= L1 for some m′ ∈ M ′ .
2
It is most remarkable that the assertions of 3.12 remain valid even without the hypothesis
that M be soluble although this played such a crucial role in the proof. Regarding the
existence assertion (1), we obtain this generalization by a routine induction on |G| : Let
40
A fundamental result on soluble groups which, however, will not be proved here, is the following:
Theorem (P. Hall 1928, 1937) A finite group G is soluble if and only if for every divisor k of |G|
such that gcd(k, |G|
k ) = 1 there exists a subgroup of order k in G. If G is soluble, any two Hall
subgroups of the same order are conjugate in G.
This theorem may be viewed as a natural generalization of Sylow’s theorem (1.11) in the universe
of all soluble finite groups. (The analogue of 1.11(2) for Hall subgroups of soluble finite groups also
holds.) Its combination of assertions on (a) existence and (b) conjugacy of subgroups with a specific
property in finite soluble groups formed the pattern for various theorems of this type which were
discovered in the 1960’s and 1970’s. In these decades, finite soluble groups were intensely studied in
this respect by group theorists all over the world, highly influenced by concepts due to W. Gaschütz.
41
This number k is called the step number of the soluble group H. It is the number of “steps” the
derived series (H(j) )j∈N0 “needs” to reach the trivial subgroup .
45
M be an arbitrary normal Hall subgroup of a finite group G. If every Sylow subgroup of
M is normal in G, M is their direct product, hence soluble by 1.8(2) and we just apply
3.12(1). If there is some non-normal Sylow subgroup P of M, we have NG (P ) < G. Now
M ∩ NG (P ) is a normal Hall subgroup of NG (P ), hence has a complement L in NG (P )
by induction. By 1.12, NG (P )M = G. It follows that L is a complement of M in G.
The conjugacy assertion (2) is unproportionally more difficult. No simple proof is
hitherto known for its general form. We prove:
3.12.1. Let M be a normal Hall subgroup of a finite group G and G/M soluble. Then
any two complements of M in G are conjugate under M.
Proof by induction on |G/M|. The case |G/M| = 1 is trivial. For the inductive step, let
M < G and L1 , L2 be complements of M in G. Consider a maximal normal subgroup
N of G such that M ≤ N. Then |G/N| = p for a prime p as G/M is soluble. Clearly,
N/M is soluble and the intersections Lj ∩ M are complements of M
N G
p
in N, hence conjugate under M by induction. Therefore it suffices to
M
∗
G
L2
assume that L1 ∩ M = L2 ∩ M. Writing D for this intersection we have
L1 DEL1 , L2 , hence DEhL1 ∪L2 i =: G∗ . Now L1 /D, L2 /D ∈ Sylp (G∗ /D),
D
∗

hence L2 = Lm
1 for some m ∈ M ∩ G by 1.11(3).
By the deep Feit-Thompson “odd order theorem” (see p. 11), either M or G/M must
be soluble if M is a normal Hall subgroup of G as at least one of the two groups must
have odd order. Thus either by 3.12(2) or by 3.12.1, all complements of M in G are
conjugate. While we gave a complete proof of the first part of the following famous result, we obtained its second part only under the additional hypothesis that M or G/M
be soluble – which, however, is satisfied by the odd order theorem:
3.13 Theorem (Schur, Zassenhaus 1937). Let G be a finite group, M a normal Hall
subgroup of G. Then
(1) M has a complement in G.
(2) Any two complements of M in G are conjugate under M.
The Schur-Zassenhaus theorem has found countless applications under the common
heading “coprime group action” which stands for lines of reasoning with group actions
involving finite groups of relatively prime orders. We give two typical examples.
3.14 Proposition. Let H be a finite subgroup of a group G, A a finite subgroup of
Aut G such that gcd(|A|, |H|) = 1. Then every A-invariant coset of H in G contains an
element which is centralized by A.
Proof. Let G∗ := A ⋉ G (see Ex. (3) in 3.1; we consider A, G as embedded in G∗ ),
x ∈ G such that (Hx)α = Hx for all α ∈ A. Then H α xα x−1 = H, hence H α = H and
−1
−1
αx ∈ αH for all α ∈ A. It follows that AH ≤ G∗ and A, Ax are complements of the
normal Hall subgroup H of AH. By 3.13(2), there exists an element h ∈ H such that
−1
Ax = Ah . Thus A is normalized by hx ∈ Hx. But NG (A) = CG (A) as G ∩ A = .
Hence hx is centralized by A.
46
3.15 Proposition. Let G be a finite group, A ≤ Aut G such that gcd(|A|, |G|) = 1.
Then for every prime p there exists an A-invariant Sylow p-subgroup of G.
Proof. Let G∗ := A ⋉ G as in the proof of 3.14, and let P ∈ Syl (G). Since G is a
normal Hall subgroup of G∗ , G ∩ NG∗ (P ) is a normal Hall subgroup of NG∗ (P ), hence
has a complement L in NG∗ (P ), by 3.13(1). By 1.12, NG∗ (P )G = G∗ so that L is a
complement of G in G∗ , hence must be conjugate to the complement A of G, by 3.13(2).
Thus A = Lx ≤ NG∗ (P )x = NG∗ (P x ) for some x ∈ G.
47
4 Transfer
We make a second application of 3.8: Let G be a group, M ≤ G, |G : M| finite,
X := G. Consider the action of G by right multiplication and the action of M by inverse
left multiplication. These group actions centralize each other, hence h̃ = idM for all
h ∈ G. The M-orbits in G are the right cosets of M in G. For any choice of a right
transversal R of M in G we have therefore, by applying both parts of 3.8:
4.0.1. wR is a homomorphism of G into M/M ′ and independent of the choice of R.42 4.1 Definition. Let G be a group, M ′ ≤ M ∗ ≤ M ≤ G, |G : M| finite, R a right
transversal of M in G. Put (M ′ m)∗ := M ∗ m for all m ∈ M. The transfer of G into
M/M ∗ is the homomorphism
vG→M/M ∗ :
G → M/M ∗ ,
g 7→ (gwR )∗ .
(If M ∗ = M ′ , the commonly used notation is vG→M instead of vG→M/M ′ .)
As M/M ∗ is abelian we will know that G 6= G′ if the image GvG→M/M ∗ is non-trivial.
The transfer will prove to be an instrument to decide if G has a nontrivial abelian factor
group. If G/G′ is finite, G′ is the intersection of the subgroups
\
G[p] := {N|G′ ≤ N ≤ G, G/N p-group}
(p prime)
so that we may confine ourselves to theTquestion if G 6= G[p] for a prime p. For an
arbitrary set π of primes we put G[π] := p∈π G[p] .
4.1.1. Let G be a finite group, p a prime and ϕ a homomorphism of G into a finite
p-group. Then ker ϕ ≥ G[p] , Gϕ = P ϕ for every P ∈ Sylp (G) 43 . In particular,
∀P ∈ Syl(G)
42
GvG→P = P vG→P ,
G/ ker vG→P ∼
= P/ ker (vG→P )|P ,
We considered the same combination of group actions in the proof of 3.6. Hence the factors in the
product definition of wR are exactly the quotients as in 3.5(1) for our current choices of group
actions. For every g ∈ G, we have in terms of the mappings ϕg

−1


Y rBg−1
Y

M ′  = (g −1 wR )−1 = gwR
Bϕg  M ′ = 
(∗)
rB g −1
B∈G /. M
B∈G /. M
as wR is a homomorphism. We may consider the wreath product U ≀ M in 3.6 as a subgroup of
ρ̄
SX ≀ M (where X = G /. M ) which has a representation as the group of monomial matrices over M0
id
(see 3.1(4)). By (∗), gwR is the product of all non-zero entries of the matrix gωΦ (ω as in 3.6, Φ as
in 3.1(4), X in place of n), modulo M ′ . We thus obtain an interpretation of gwR as the “signless
determinant modulo M ′ ” of the monomial matrix associated with g.
43
More generally, Gϕ = P ϕ holds for every subset P of G such that P G[p] = G.
48
an obvious consequence of the homomorphism theorem.
Given g ∈ G, we may exploit the fact that wR is independent of the choice of R for the
calculation of gvG→M/M ∗ , choosing R in dependence of g. This is the main idea behind
the following important remark:
4.1.2 (The transfer formula). Let g ∈ G, B be the set of hgi-orbits in G /. M. For every
B ∈ B let xB ∈ G such that MxB ∈ B. Then
Y
x−1
g |B| B ,
gvG→M = M ′
B∈B
P
B∈B
|B| = |G : M|, and |B|o(g) for all B ∈ B if o(g) is finite.
Proof. If B ∈ B, then B = {MxB , MxB g, . . . , MxB g |B|−1 }, MxB g |B| = MxB . By 1.4.1,
Ṡ
S
B = G /. M so P
that R := B∈B {xB , xB g, . . . , xB g |B|−1} is a right transversal of M in
G; in particular, B∈B |B| = |G : M|. It follows that
gwR = M ′
Y
B∈B
Y xB
xB xB g
xB g |B|−1 ′
·
·
·
,
=
M
xB g |B| xB g
xB g |B|−1
xB g |B|
B∈B
by 3.4.1(2). For every B ∈ B, the element m ∈ M with the property m−1 xB g |B| = xB
equals xB g |B| x−1
B , which proves the formula. The last assertion is clear by 1.7(2).
Consequently, if hgiy ∩ M ⊆ M ∗ for all y ∈ G, then g ∈ ker vG→M/M ∗ . In the following
remark, we consider the image of an element of M under the transfer of G into M/M ∗ ,
choosing M ∗ so that it contains a certain set of commutators:
4.1.3. Let m ∈ M, {m̃−1 m′ |m̃ ∈ hmi, ∃x ∈ G m′ = m̃x ∈ M} ⊆ M ∗ . Then
mvG→M/M ∗ = M ∗ m|G:M | .
Proof. By hypothesis, m̃x ∈ M ∗ m̃ if m̃ ∈ hmi and x ∈ G such that m̃x ∈ M. By 4.1.2,
Y
Y
x−1
m|B| B = M ∗
m|B| = M ∗ m|G:M | .
mvG→M/M ∗ = M ∗
B∈B
B∈B
4.1.4. Let G be finite, F := hm−1 m′ |m, m′ ∈ M, ∃x ∈ G m′ = mx i. Then
F ≤ M ∩ G′ ≤ ker (vG→M )|M ≤ ker (vG→M/F )|M .
If gcd(|M/F |, |G : M|) = 1, equality holds throughout this subgroup chain.
Proof. The first assertion consists of trivial consequences of the definition of F . By
4.1.3, F m|G:M | = F for all m ∈ M ∩ ker vG→M/F , hence m ∈ F if |M/F |, |G : M| are
coprime.
49
4.2 Lemma (Focal subgroup lemma). Let G be a finite group, p a prime, P ∈ Sylp (G).
(1) hm−1 m′ |m, m′ ∈ P, ∃x ∈ G
m′ = mx i = P ∩ G′ = ker (vG→P )|P ,
(2) ker vG→P = G[p] .
Proof. (1) follows from 4.1.4 (with M := P ). Furthermore, G[p] ≤ ker vG→P by 4.1.1.
Since P G[p] = G and P ∩ G′ = P ∩ G[p] , the claim in (2) follows.
The subgroup P ∩G′ is called the focal subgroup of P . By 4.2, P/ ker (vG→P )|P ∼
= G/G[p].
Hence the transfer of G into P , restricted to P , determines if G[p] and G coincide or not.
Trivially we have
[P, NG (P )] ≤ P ∩ G′
and P ∩ Q′ ≤ P ∩ G′ for all Q ∈ Sylp (G).
An important refinement of 4.2(1) is the result that these subgroups indeed generate the
focal subgroup of P . We prepare the proof of this result by an extension of the transfer
formula 4.1.2 for the case of an element g of prime power order. By 4.2(1), it suffices
to treat this case in order to determine the focal subgroup of P . We will exploit the
fact that the subgroups of hgi then form a chain. The extension of 4.1.2 is based on the
observation that any subgroup H of G such that g ∈ H gives rise to a partition of the
set of all hgi-orbits in G /. M (see 4.1.2): Clearly, for every y ∈ G, the H-orbit of My in
G /. M splits into hgi-orbits. Their lengths are divisors of o(g)(= |hgi|), by 1.7(2).
4.3 Proposition. Let G be a group, M ≤ G, |G : M| finite, g ∈ H ≤ G, o(g) a prime
power, B the set of H-orbits in G /. M. For every B ∈ B choose xB ∈ G with MxB ∈ B
such that the hgi-orbit of MxB is of minimal length. Let M ′ hM ∩H ′y |y ∈ Gi ≤ M ∗ ≤ M.
Then
Y
−1
gvG→M/M ∗ = M ∗
(g |B| )xB ,
B∈B
(g |B| )
x−1
B
∈ M, |B| = 1 if and only if H ≤ M xB .
Ṡ
Proof. For every B ∈ B choose RB ⊆ H such that MxB H = r∈RB MxB rhgi, w. l. o. g.
−1
∈ N, (g j )(xB r) ∈ M}. This
1G ∈ RB . For every r ∈ RB put |B|r := min{j|j
is
P
the length of the hgi-orbit of MxB r, hence r∈RB |B|r = |B|. We have |B|1G |B|r
for all r ∈ RB by the choice of xB and the fact that o(g) is a prime power, thus
Q
−1
−1
−1
−1
(g |B|r )xB ∈ h(g |B|1G )xB i ≤ M. We conclude that (g |B| )xB = r∈RB (g |B|r )xB ∈ M and
(g |B|r )r
−1 x−1
B
−1
−1
−1
(g −|B|r )xB = [r −1 , g −|B|r ]xB ∈ M ∩ H ′ xB ≤ M ∗ .
Making use of 4.1.2, this implies
Y
Y Y
Y Y
−1
−1
−1
(g |B|r )xB = M ∗
(g |B| )xB .
(g |B|r )(xB r) = M ∗
gvG→M/M ∗ = M ∗
B∈B r∈RB
B∈B r∈RB
Finally, |B| = 1 ⇔ MxB H = MxB ⇔ M xB H = M xB ⇔ H ≤ M xB .
50
B∈B
If H = hgi, 4.3 reduces to 4.1.2. If H = G we obtain: gvG→M/(M ∩G′ ) = (M ∩ G′ )g |G:M |
for all g of prime power order, hence for all g ∈ G; this is also a simple consequence of
4.1.2. The proof of 4.4 will make use of the case H = M ∈ Syl(G):
4.4 Theorem (Grün 1935). Let G be a finite group, p a prime, P ∈ Sylp (G). Then
P ∩ G′ = hP ∩ Q′ |Q ∈ Sylp (G)i [P, NG (P )].
Proof. Put P ∗ := hP ∩ Q′ |Q ∈ Sylp (G)i. We show, by induction on o(g), that every
g ∈ P ∩ G′ is contained in P ∗ [P, NG (P )]. This is trivial for g = 1G . For the inductive
step let g ∈ P ∩ G′ and o(g) > 1. By 4.3, we have
Y
−1
−1
P ∗ = gvG→P/P ∗ = P ∗ (g |B| )xB , (g |B| )xB ∈ P ∩ G′ for all P -orbits B in G /. P.
B
−1
−1
If |B| > 1, then o((g |B| )xB ) < o(g), hence inductively (g |B| )xB ∈ P ∗ [P, NG (P )]. As
|B| = 1 if and only if xB ∈ NG (P ), it follows that
Y
Y −1 Y
−1
g |NG (P ):P | ≡
g[g, x−1
]
=
g xB ≡
(g |B| )xB ≡ 1G mod P ∗[P, NG (P )],
B
B∈NG (P )/P
B
|B|=1
hence g ∈ P ∗ [P, NG (P )] because p ∤ |NG (P ) : P |.
By 4.4, two different aspects of the embedding of the Sylow subgroup P in G allow
to determine the structure of G/G[p] : The intersections of P with the commutator
subgroups of all conjugates of P and the largest quotient of P which is centralized by
NG (P ). Given any p-group, one can now study the different possible cases which may
arise – in dependence of the subgroup structure of P and the p′ -subgroups of Aut G – if
the given group is to occur as a Sylow p-subgroup of a finite group. For example, in [G],
7.7, there is an analysis of groups with dihedral Sylow 2-subgroups along these lines.
The complexity which may be hidden in the two above-mentioned aspects is avoided by
brute force if the hypotheses are chosen such that both factors of P ∩G′ in 4.4 are trivial.
This is the case in one of the oldest results in the spirit of this chapter, by Burnside
(1900): Let p be prime, G a finite group with an abelian Sylow p-subgroup P . Suppose
that NG (P ) = CG (P ). Then P is a complement of G[p] in G. On the basis of 4.4, the
proof is trivial because we immediately obtain P ∩ G′ = .44 We will give an extended
version of Burnside’s result in 5.16. Burnside’s hypothesis is clearly satisfied if p is the
smallest prime divisor of |G| and P is cyclic, because each prime divisor 6= p of |Aut P |
then divides p − 1. Thus we observe:
4.5 Corollary. Let G be a finite group with a cyclic Sylow p-subgroup P where p is the
smallest prime divisor of |G|. Then P is a complement of G[p] in G.
44
We add a direct proof, using only 4.2: Let m ∈ P , x ∈ G such that mx ∈ P . By 4.2(1), it suffices
−1
to show that mx = m. As m ∈ P ∩ P x and the Sylow p-subgroups of G are abelian, we have
−1
x−1
x−1
P, P
≤ CG (m). Clearly then P, P
∈ Sylp (CG (m)), hence P x = P z for some z ∈ CG (m), by
1.11(3). It follows that zx ∈ NG (P ) = CG (P ), by hypothesis. Hence x ∈ z −1 CG (P ) ⊆ CG (m). 51
Under appropriate hypotheses it may be proved that G and a subgroup G̃ containing
a Sylow p-subgroup P of G have isomorphic largest abelian p-factor groups.45 In this
direction we observe the following simple consequence of 4.4:
4.6 Corollary. Let G be a finite group, p a prime, Z ⊆ P ∈ Sylp (G), G̃ := NG (Z).
char
Suppose that P ∩ Q′ ≤ G̃ ′ for all Q ∈ Sylp G. Then G/G[p] ∼
= G̃/G̃[p] .
Proof. We have to show that P ∩ G′ = P ∩ G̃ ′ . For the nontrivial inclusion we just
have to remark that NG (P ) ≤ G̃, hence [P, NG (P )] ≤ G̃ ′ . The claim then follows from
4.4.
By nothing else than making clever use of 1.11, Grün showed that the hypothesis of 4.6
is satisfied with Z = Z(P ) if for all x ∈ G either Z(P )x = Z(P ) or Z(P )x 6⊆ P . This
result is known as the “2nd theorem by Grün” (see [H], IV, 3.7 or [R], 10.2.3 for a proof
which is related to the argument given in footnote 44; furthermore, cf. [KS], 7.1.8).
For a 3rd application of 3.8 let G be a group, M ≤ N E G, |N : M| finite, X := N.
Let M act by inverse left multiplication λ, G by conjugation κ on N. For any g ∈ G,
m ∈ M, x ∈ N we have
−1 g
x(mλ)gκ = m−1 xg
= (mg )−1 x,
hence Mλ is normalized by gκ if and only if M g ⊆ M. We conclude that the action of
NG (M) on N given by conjugation normalizes the action of M on N given by λ. Let R
be a right transversal of M in N and wR : NG (M) → M/M ′ the crossed homomorphism
from 3.8. We observe
4.6.1. ∀m ∈ M
mwR = (M ′ m−1 )|N :M | mvN →M .
Proof. Let m ∈ M. Making use of 3.7.1(II), we have
mwR =
|N :M |
R
R
Rm
R
=
=
= M ′ m−1
mvN →M ,
m
R
(Rm)(mλ)
(Rm)(mλ) Rm
because all M-orbits in N /. M are of length 1, and
s
s(mλ)
= m−1 for all s ∈ N.
4.7 Lemma. Let G be a group, M ′ ≤ M ∗ ≤ M ≤ N E G, N finite. Suppose that
M ∗ vN →M ⊆ M ∗ /M ′ . For all m ∈ M set m̄ := M ∗ m, and define ψ, ω ∈ End M̄ by
j
j+1
m̄ψ := m̄−|N :M | , m̄ω := mvN →M̄ . Put ϕ := ψ ∔ ω and let j ∈ N such that M̄ ϕ = M̄ ϕ .
j
Then M̄ ϕ is a complemented normal subgroup of (NG (M) ∩ NG (M ∗ ))/M ∗ .
45
A subgroup G̃ with this property is said to control p-transfer of G. For a striking consequence of this
see 4.10. Frequently G̃ := NG (P ) controls p-transfer of G, due to the following result:
Theorem (Yoshida 1978) Let G be a finite group, p a prime, P ∈ Sylp (G). Suppose that P has no
factor group isomorphic to Cp ≀ Cp . Then NG (P ) controls p-transfer of G.
For a proof, see [I], ch. 10.
52
G
D
N
M
j
M∗

U : U/M ∗ = M̄ ϕ
Proof. Let D := NG (M) ∩ NG (M ∗ ). By 4.6.1 there exists a
crossed homomorphism w of D into M̄ such that mw = m̄ϕ
for all m ∈ M . From this it is easily seen that wϕi is a crossed
homomorphism of D for all i ∈ N0 . By the choice of j we have
j
j+1
j
j
M̄ ϕ = M̄ ϕ ≤ D(wϕj ) ≤ M̄ ϕ . Hence D(wϕj ) = M̄ ϕ is norj
malized by D, by 3.3.4, and wϕj induces a permutation on M̄ ϕ .
j
By 3.4(4), ker(wϕj )/M ∗ is a complement of M̄ ϕ in D/M ∗ .
The hypotheses of 4.7 become considerably simpler if M is contained in ker vN →M̄ . The
following corollaries deal with situations in which this is the case:
4.8 Corollary. Let G be a group, M ≤ N E G, N finite. Let π be a set of primes which
do not divide |N : (M ∩ N ′ )|. Then M/M [π] has a complement in NG (M)/M [π] .
Proof. Let M̄ := M/M [π] . Then vN →M̄ is trivial so that ϕ equals ψ (in the notation of
4.7) and is an automorphism of M̄ . By 4.7 (with j = 1), the claim follows.
For every finite group G and prime p we write O p (G) for the smallest normal subgroup
of G the quotient of which is a p-group. Clearly, the Sylow p-subgroups of O p (G) are
contained in O p (G)′ . Thus 4.8 implies:
4.9 Corollary. Let G be a finite group, p a prime, Q ∈ Sylp (O p (G)).
G
O p (G)
(1) If P ∈ Sylp (G) such that Q ≤ P , there exists a subgroup S of P such
that QS = P , Q ∩ S = Q′ .
P
Q
Q′
(2) (Gaschütz 1952) If Q is abelian, then O p (G) has a complement in G.

While 4.9(2) is an immediate consequence of 4.9(1), the following application of 4.9(1)
is a little less obvious. For a long time, the only known proofs used cohomology or
character theory. We thank M. I. Isaacs for friendly calling our attention to the proof in
[GI] which is based on transfer methods. A comparison with the approach given here is
interesting because the ideas overlap but the strategies are different.
4.10 Theorem (Tate 1964). Let G be a finite group, p a prime, P ∈ Sylp (G), P ≤ G̃ ≤ G.
Then
G/G[p] ∼
= G̃/O p(G̃).
= G̃/G̃[p] ⇔ G/O p (G) ∼
Proof. The isomorphism on the left is clearly implied by the isomorphism on the
right and means that G̃[p] = G̃ ∩ G[p] . We now assume this equation and show that
O p (G̃) = G̃ ∩ O p (G).
Set Q := P ∩ O p (G), Q̃ = P ∩ O p (G̃), F := P ∩ G′ . Then
F = P ∩ G[p] = P ∩ G̃[p] , F/Q = (P/Q)′ , F/Q̃ = (P/Q̃)′ .
By 4.9(1), there exists a subgroup S of P such that SQ = P ,
S ∩ Q = Q′ . By 1.8.2, the assumption Q̃ < Q would imply
Q̃Q′ < Q, hence Q̃Q′ S < P . But then Q̃(≤ Q̃Q′ S) would
be contained in a maximal subgroup of P not containing F ,
in contradiction to 1.8.2 as F/Q̃ = (P/Q̃)′ . It follows that
Q̃ = Q, i. e., O p (G̃) = G̃ ∩ O p (G).
53
O p (G)
G[p]
G
G̃
O p (G̃)
Q
F
P
Q̃Q′
Q̃

5 Nilpotency
The concept of nilpotency makes extensive use of group commutators (see p. 23). Therefore we recall a number of their important properties. The definition and a first variation give, for arbitrary elements x, y of a group G: [x, y] := x−1 y −1xy = (y −1x−1 yx)−1 =
[y, x]−1 . For arbitrary subsets U, V of G we conclude that [U, V ] (the subgroup generated
by all [x, y] where x ∈ U, y ∈ V i) equals [V, U].
[xy, z] = [x, z]y [y, z] = [x, z][[x, z], y][y, z]
5.0.1. ∀x, y, z ∈ G
The first equation in 5.0.1 means that, for every z ∈ G, the mapping [. , z] : G → G,
x 7→ [x, z], is a crossed homomorphism. Moreover, it implies
∀x, y, z ∈ G [z, xy] = [xy, z]−1 = [y, z]−1 ([x, z]−1 )y = [z, y][z, x]y = [z, y][z, x][[z, x], y],
furthermore
5.0.2. ∀U, V ⊆ G
[U, V ] E hU ∪ V i.
5.0.3. [U, V ] ⊆ U ⇔ V ≤ NG (U) for all U, V ≤ G. In particular, [U, G] ⊆ U ⇔ U E G.
Proof. [U, V ] ⊆ U ⇔ ∀x ∈ U ∀y ∈ V x−1 y −1 xy ∈ U ⇔ ∀x ∈ U ∀y ∈ V
V ⊆ NG (U). The last assertion is the special case V = G.
y −1 xy ∈ U ⇔
5.0.4. Let Ω be a set of operators on G, U, V ≤ G. Then [U, V ] ≤ G,
Ω
Ω
because [x, y]α = [xα , y α] for all x, y ∈ G, α ∈ End G. Thus U α ⊆ U, V α ⊆ V implies
[U, V ]α ⊆ [U, V ].
5.0.4 means that [ . , . ] induces an operation on the set of all Ω- subgroups of G. Choosing
Ω = G, Ω = Aut G resp., we obtain as special cases
U, V E G ⇒ [U, V ] E G;
U, V ≤ G ⇒ [U, V ] ≤ G.
char
char
Furthermore, 5.0.1 implies
5.0.5. ∀U, V, W E G
[U, W ][V, W ] = [UV, W ] = [W, UV ] = [W, U][W, V ]
5.0.6. Let U ≤ V ≤ G. Then [V, G] ⊆ U if and only if U, V E G and V /U ≤ Z(G/U).
Proof. If [V, G] ⊆ U, then [U, G] ⊆ [V, G] ⊆ U ⊆ V , hence U, V E G by 5.0.3. Thus
[V, G] E G by 5.0.4 and v −1 v g ∈ [V, G] ⊆ U, hence vU = v g U for all v ∈ V , g ∈ G, i. e.,
V /U ≤ Z(G/U). Conversely, the latter implies that vU = v g U for all v ∈ V , g ∈ G,
hence [V, G] ⊆ U.
54
5.1 Definition. Let G be a group. A finite chain K of normal subgroups of G is called
G-central if each factor in K (see 2.13) is centralized by G. If U is the smallest, V the
largest element of K, then K is called an (ascending) chain from U to V or a (descending)
chain from V to U.
1st special case: Put Z0 (G) = . Inductively, having defined the normal subgroup
Zj−1 of G for some j ∈ N, let Zj (G) be the subgroup of G such that Zj−1 (G) ≤ Zj (G)
and Zj (G)/Zj−1(G) = Z(G/Zj−1(G)). The set {Zj (G)|j ∈ N0 } is called the ascending
central series of G. Its union46
[
Zj (G)
Z∞ (G) :=
j∈N0
is called the hypercentre of G. If there exists an n ∈ N0 such that Zn (G) = Zn+1 (G),
then Zn (G) = Zn+j (G) = Z∞ (G) for all j ∈ N. Then {Zj (G)|j ∈ n ∪ {0}} is a G-central
chain from  to Z∞ (G).
2nd special case: Put K0 (G) := G. Inductively, having defined the normal subgroup
Kj−1 (G) of G for some j ∈ N, let Kj (G) := [Kj−1 (G), G]. The set {Kj (G)|j ∈ N0 } is
called the descending central series of G.47 We set
\
Kj (G).
K∞ (G) :=
j∈N0
If there exists an n ∈ N0 such that Kn (G) = Kn+1 (G), then Kn (G) = Kn+j (G) = K∞ (G)
for all j ∈ N. Then, by 5.0.6, {Kj (G)|j ∈ n∪{0}} is a G-central chain from G to K∞ (G).
The group G is called nilpotent if there exists a G-central chain from  to G. If K is such
a chain, then, in particular, every element of K is abelian. Thus we observe:
5.1.1. Every nilpotent group is soluble.
5.1.2. Let n ∈ N, j ∈ n. Then Kj−1 (G) ≤ Zn−(j−1) (G) ⇔ Kj (G) ≤ Zn−j (G).
Proof. If Kj−1 (G) ≤ Zn−(j−1) (G), then Kj (G) = [Kj−1 (G), G] ≤ [Zn−j+1(G), G] ≤
Zn−j (G), by 5.0.6. If Kj (G) ≤ Zn−j (G), then [Kj−1 (G), G] ≤ Zn−j (G), hence Kj−1 (G) ≤
Zn−j+1(G) by 5.0.6.
Consequently, Kn (G) =  if and only if Zn (G) = G which may be seen by repeated
application of 5.1.2:
Kn (G) =  ⇔ Kn (G) ≤ Z0 (G) ⇔ Kn−1 (G) ≤ Z1 (G) ⇔ · · ·
⇔ K0 (G) ≤ Zn (G).
5.1.3. Let {H0 , . . . , Hn } be a G-central chain such that G = H0 ≥ · · · ≥ Hn . Then
Kj (G) ≤ Hj for all j ∈ n ∪ {0}.
46
More precisely, the notation should be Zω (G) where ω is the first limit ordinal. But we will not dwell
on central series in this more general sense here.
47
A traditional notation for its members is obtained by putting γj (G) := Kj−1 (G) for all j ∈ N. This
shift of subscripts would not be adequate for our present purposes. In other contexts, however, it is
reasonable and therefore frequently seen in the literature.
55
Proof. The claim is trivial for j = 0. If j > 0 and Kj−1 (G) ≤ Hj−1 , then Kj =
[Kj−1 (G), G] ≤ [Hj−1 , G] ≤ Hj , by 5.0.6.
5.1.4. The following conditions are equivalent:
(i) G is nilpotent,
(ii) ∃m ∈ N0
Zm (G) = G,
(iii) ∃n ∈ N0
Kn (G) = .
Proof. By 5.1.3, (iii) implies (i), and the converse is trivial. The equivalence of (ii) and
(iii) follows from the observed (stronger) consequence of 5.1.2.
If G is nilpotent, we conclude the following from what we have seen after 5.1.2:
min{n|n ∈ N0 , Zn (G) = G} = min{n|n ∈ N0 , Kn (G) = },
i. e., the ascending central chain and the descending central chain of G have the same
length. This length is called the nilpotency class of G, denoted by cl(G). From 5.1.3 we
conclude
5.1.5. If K is a G-central chain from G to , then cl(G) ≤ l(K).
Nilpotency has certain closure properties which hold trivially but are most useful. Applying 2.13.2 where K is a G-central chain from G to , Ω = H, = G resp., we obtain
5.1.6. Subgroups and factor groups of nilpotent groups are nilpotent.
From the definition of the ascending central chain and 5.1.6 we obtain
5.1.7. G is nilpotent if and only if G/Z(G) is nilpotent.
5.1.8. Let X be a set of nilpotent groups. Then × X is nilpotent if and only if there
exists an n ∈ N such that cl(G) ≤ n for all G ∈ X ,
because Zj (× X ) =
× Zj (G) for all j ∈ N.
G∈X
5.1.9.TLet Y be a set of normal subgroups of G such that G/N is nilpotent for all N ∈ Y,
D := Y. If there exists an n ∈ N such that cl(G) ≤ n for all G ∈ Y, then G/D is
nilpotent and cl(G/D) ≤ n,
because Kn (G) ≤ N for all N ∈ Y, hence Kn (G) ≤ D, Kn (G/D) = .
5.1.10.
T Let Y be the set of all normal subgroups N of G such that G/N is nilpotent.
Then Y = K∞ (G).
T
Proof. Clearly K∞ (G) ≤ N for all N ∈ Y. On the other hand, K∞ (G) = n∈N Kn (G)
and Kn (G) ∈ Y for all n ∈ N.
56
T
The
Tgroup Y in 5.1.10 is called the nilpotent residual of G. If G is finite, the quotient
of Y is nilpotent by 5.1.9. Thus every finite group has a smallest normal subgroup
the quotient of which is nilpotent, namely its nilpotent residual.48
5.2 Proposition. Let G be a nilpotent group.
(1) U < G ⇒ U < NG (U),
(2) U < G ⇒ U ⊳ G.
max
G
Proof. (1) Let n ∈ N be minimal such that Zn (G) 6≤ U. Then
[U, Zn (G)] ≤ [G, Zn (G)] ≤ Zn−1 (G) ≤ U,
hence Zn (G) ≤ NG (U) by 5.0.3, U < UZn (G) ≤ NG (U). (2) is an
immediate consequence of (1).
U
Zn (G)
Zn−1 (G)
Let p be a prime and G a p-group. It is easily seen by induction
on |G| that G is nilpoT
49
p
tent. For an arbitrary finite group
T G thisp implies that G/ p prime O (G) is nilpotent, by
5.1.9. In particular, K∞ (G) ≤ p prime O (G). We shall soon see that, in fact, equality
holds.
5.3 Main Theorem for finite nilpotent groups. For every finite group G the following are equivalent50 :
(i) G is nilpotent
(ii) ∀U < G U < NG (U)
(iii) ∀U ≤ G
U EE G
(iv) ∀U < G
U ⊳G
max
(v) ∀P ∈ Syl(G) P E G
(vi) ∀P ∈ Syl(G) P ≤ G
char
(vii) Syl(G) is a direct decomposition of G.
Proof. (i)⇒(ii) holds by 5.2(1).
(ii)⇒(iii) is proved by induction on |G : U|: The case |G : U| = 1 is trivial. Now let
U < G. By hypothesis, U < NG (U). Inductively, NG (U) EE G so that U is subnormal
in G.
48
Note that, despite the name, thisTterm does not refer to the structure of K∞ (G), and for infinite
groups not even the quotient of Y is necessarily nilpotent.
49
For the inductive step, let G 6= . Then Z(G) 6=  by 1.8(2). If we assume inductively that the
p-group G/Z(G) is nilpotent, it follows that G is nilpotent by 5.1.7.
50
A further equivalence will be obtained in 5.12(2).
57
(iii)⇒(iv) is trivial.
(iv)⇒(v): Assume that P 5 G for some P ∈ Syl(G). Then NG (P ) < G so that there
exists a maximal subgroup U of G such that NG (P ) ≤ U. By hypothesis, U E G. Now
1.12 implies U = NG (P )U = G, a contradiction.
(v)⇒(vi) If α ∈ Syl(G), P ∈ Syl(G), then P α ∈ Syl(G). By 1.11(3), there exists an
element g ∈ G such that P α = P g . By hypothesis, this means P α = P .
(vi)⇒(vii) By hypothesis, Syl(G) is a set of normalQsubgroups ofQmutually coprime
orders. Hence their product is a direct product and | Syl(G)| = P ∈Syl(G) |P | = |G|,
Q
implying that Syl(G) = G.
(vii)⇒(i) follows from 5.1.8 as p-groups are nilpotent.
T
5.4 Corollary. K∞ (G) = p prime O p (G) for every finite group G.
Proof. Let G be T
a finite group. We proved one inclusion before 5.3 so that it suffices to
show K∞ (G) ≥ p prime O p (G). The quotient Ḡ := G/K∞ (G) is nilpotent (see 5.1.10),
hence the direct product of its Sylow subgroups, by 5.3. It follows T
that O p (Ḡ) is the
product of all Q ∈ Syl(Ḡ) such that p ∤ |Q|, for all primes p. Therefore p prime O p (Ḡ) = .
The claim follows.
5.5 Corollary. If G is a finite nilpotent group, then Aut G ∼
= ×P ∈Syl(G) Aut P .
Proof. Let πP be the projection of G onto P ∈ Syl(G) with respect to the direct decomposition
Q Syl(G) of G (cf. 5.3). Let αP ∈ Aut P for every P ∈ Syl(G). Then α : G → G,
gQ7→ P (g πP )αP , is an automorphism of G, and the mapping ∆ : ×P Aut P → Aut G,
P αP 7→ α, is a monomorphism. As every P ∈ Syl(G) is characteristic by 5.3, every
automorphism α of G normalizes every
Q Sylow subgroup P of G and induces an automorphism αP on P . It follows that ( P αP )∆ = α so that ∆ is an isomorphism.
The commutator subgroup of a group G is a particular case of a subgroup of type
[U, V ] where U, V ⊆ G. The same holds, more generally, for every member Kn (G) of
the descending central series of G if we define inductively for any sequence (Un )n∈N of
subsets of G the left-normed commutator subgroup of (U1 , . . . , Un+1 ) by
∀n ∈ N [U1 , . . . , Un+1 ] := [[U1 , . . . , Un ], Un+1 ].
Furthermore, we use the same recursive definition for elements of G in place of subsets.
It is useful to define, moreover, [U] := hUi for U ⊆ G, [g] := g for g ∈ G. From 5.0.3 we
conclude by a trivial induction
5.5.1. Let Ui ⊆ N E G for j mutually distinct i ∈ n. Then [U1 , . . . , Un ] ≤ Kj−1 (N). Here, by putting K−1 (N) = G, also the case j = 0 is allowed.
5.5.2. Let M, N E G, n ∈ N. Then
[MN, . . ., MN] =
n
Y
(U1 ,...,Un )∈{M,N }n
n
Y
[U1 , . . . , Un ] ≤
(Kj−1 (M) ∩ Kn−j−1(N)).
j=0
58
Proof. The claimed equality is clear for n = 1.
Q Inductively we conclude, using 5.0.5,
[MN, . . . , MN] = [W, MN] = [W, M][W, N] = (U1 ,...,Un+1 )∈{M,N }n+1 [U1 , . . . , Un+1 ] where
n+1
Q
W = (U1 ,...,Un )∈{M,N }n [U1 , . . . , Un ]. The claimed inequality follows from 5.5.1.
5.6 Corollary (Fitting 1938). Let M, N be nilpotent normal subgroups of a group G.
Then MN is a nilpotent normal subgroup of G, and cl(MN) ≤ cl(M) + cl(N).
Proof. Put n := cl(M) + cl(N) + 1. Then Kj−1 (N) =  or Kn−j−1 (M) =  for all j such
that 0 ≤ j ≤ n. Thus 5.5.2 implies Kn (MN) = . The claim follows.
5.7 Definition. Let G be a group. Then
Y
F (G) :=
N
NEG
N nilpotent
is called the Fitting subgroup of G. The Fitting subgroup may be viewed as a dualization
of the nilpotent residual, where intersections of normal subgroups with nilpotent quotient
are replaced by products of nilpotent normal subgroups. Clearly,
5.7.1. F (G) ≤ G.
char
5.7.2. Z∞ (G) ≤ F (G),
because for every n ∈ N, Zn (G) is a nilpotent normal subgroup of G: {Zj (G)|j ∈ n} is a
finite G-central chain, a fortiori a finite Zn (G)-central chain from  to Zn (G). It follows
that Zn (G) ≤ F (G), hence the claim.
The tiny example G = S3 shows already that, in general, Z∞ (G) is properly contained
in F (G). We have Z∞ (S3 ) =  < A3 = F (S3 ).
Without any restriction on G, the hypercentre Z∞ (G), a fortiori F (G), need not be
nilpotent, as may be seen from 5.1.8. But 5.6 is generalized by a simple induction to
5.7.3. The product of finitely many nilpotent normal subgroups of G is nilpotent. In
particular, F (G) is nilpotent if G is finite.
Hence every finite group has a largest nilpotent normal subgroup, namely, its Fitting
subgroup.
5.7.4. Let G be finite. Then ∀N E G F (G) ∩ N = F (N).
Proof. As F (N) ≤ N E G, F (N) is a nilpotent normal subgroup of G. Furthermore,
char
F (G) ∩ N is a nilpotent normal subgroup of N. The claim follows.
5.7.5. Let G be a group, p a prime, M, N E G, M, N p-groups. Then MN E G and MN
is a p-group,
59
because |MN| =
|M ||N |
|M ∩N |
is a power of p.
We set
Op (G) :=
Y
N.
NEG
N p-group
Q
As every p-group is nilpotent, we have p prime Op (G) ≤ F (G).
Q
5.7.6. Let G be finite. Then p prime Op (G) = F (G).
Proof. F (G) is nilpotent by 5.7.3. For all primes p, O
Qp (F (G)) is its (by
Q 5.3 unique)
Sylow p-subgroup. Hence Op (F (G)) ≤ Op (G), F (G) = p Op (F (G)) ≤ p Op (G).
It is not difficult to see that the left-normed commutators [x1 , . . . , xn+1 ] (xi ∈ G) generate
the group Kn (G), for every n ∈ N0 . As a consequence we have the following equivalence
for every n ∈ N0 :
G is nilpotent and cl(G) ≤ n
⇔
∀x1 , . . . , xn+1 ∈ G [x1 , . . . , xn+1 ] = 1G .
It is remarkable that the property of being nilpotent is for finite groups already implied
by a condition of this kind in which only two variables for group elements occur; this
was discovered by M. Zorn. There is a strong analogy with a basic result in the theory of
Lie algebras from where also the terminology (of “Engel elements”) has been adopted:
An element g of a group G is called 51
right Engel if:
∀x ∈ G ∃n ∈ N [g, x . . . , x] = 1G ,
| {z }
n
left Engel if:
∀x ∈ G ∃n ∈ N [x, g, . . . , g ] = 1G .
| {z }
n
A right Engel element is thus an element which is “treated nilpotently” by all mappings
[ . , x] (x ∈ G) while a left Engel element is characterized by the property that the
mapping [ . , g] is nil, i. e., “acts nilpotently” on every x ∈ G. An Engel element is a
group element which is right or left Engel. The main result about Engel elements for
finite52 groups is the following:
Theorem (Baer 1957) Let G be a finite group, g ∈ G.
(1) g is right Engel if and only if g ∈ Z∞ (G),
(2) g is left Engel if and only if g ∈ F (G).
51
52
Huppert exchanges the meaning of a left and a right Engel element in his definition ([H], §6, 6.12).
More generally, Baer [B] considers group elements the normal subgroup closure of which is noetherian,
i. e., all of its subgroups are finitely generated. In particular, his results hold for noetherian groups.
60
Consequently, every right Engel element of a finite 53 group is left Engel, by 5.7.2. It is
still unknown if this statement is true for arbitrary groups.54 A main step in the proof
of Baer’s theorem is to show that a finite group is nilpotent if it has a set of generators
which are Engel elements. A nontrivial part consists in the preliminary result that such
a group must be soluble. A detailed proof may be found in [H], III, §6. (But be aware
of footnote 51!) Engel conditions in infinite groups have been intensely, but still not
conclusively discussed in numerous papers.
The Fitting subgroup has a major impact on the study of the structure of a finite soluble
group because of the following result:
5.8 Lemma. Let G be a finite soluble group. Then CG (F (G)) ≤ F (G).
G
F (G)
M
D
C
Proof. Let C := CG (F (G)), D := F (G) ∩ C(= Z(F (G))). By 1.4.8,
C E G. Being contained in F (G), D is centralized by C, hence
D ≤ Z(C). Assume that C 6= D. Then let D < M ≤ C such
that M/D E G/D. As G/D is soluble, M/D is elementary abelian
min
(2.12). Hence Z2 (M) = M so that M is nilpotent. Thus M ≤ F (G),
a contradiction. Therefore C = D ≤ F (G).
For any normal subgroup N of a group G, G/CG (N) is isomorphic to a subgroup of
Aut G, by 1.4.10, and Z(N) = N ∩ CG (N). In the special case of the Fitting subgroup
of a finite group G we obtain, more specifically, by 5.5 and 5.8:
5.8.1.
Let G be a finite group. Then G/CG (F (G)) is isomorphic to a subgroup of
Q
Aut
P where P ranges over Syl(F (G)). If G is soluble, CG (F (G)) = Z(F (G)). P
Consequently, if G is a finite soluble group and P1 , . . . , Pr are the Sylow subgroups 6= 
of F (G), D := Z(F (G)), then
• D is an abelian normal subgroup of G isomorphic to Z(P1 ) × · · · × Z(Pr ),
• G/D is isomorphic to a subgroup of Aut P1 × · · · × Aut Pr .
Thus every finite soluble group G, viewed as an extension of D by G/D, is composed by
constituents which may be structurally derived from p-groups, namely from the Sylow
subgroups of F (G). For a given finite set of p-groups we have obtained a “structural
bound” for the possible finite soluble groups having their direct product as its Fitting
subgroup. More generally, such a structural restriction holds for all groups G with the
property that CG (F (G)) ≤ F (G). For this reason, groups with this property are called
constrained.
53
54
more generally: noetherian, see footnote 52
−1
−1
The transformation [x, g, . . . , g] = [g −1 , g x , . . ., g x ]xg (which is a simple consequence of 5.0.1)
n+1
n
shows that the inverse of a right Engel element h is always left Engel. Note that the hypothesis is
not fully used here as it suffices that h(= g −1 ) be treated nilpotently by all conjugates of h−1 .
61
5.9 Definition. Let G be a group. Then
Φ(G) :=
\
H
H < G
max
or H=G
is called the Frattini subgroup of G. For example, Φ(D8 ) = Z(D8 ) = D8′ is of order 2,55
Φ(S3 ) = , Φ(Cpn ) is the unique maximal subgroup of Cpn , Φ(Q, +) = Q as the group
(Q, +) has no maximal subgroup.56
5.9.1. Let p be a prime and G abelian such that g p = 1G for all g ∈ G. Then Φ(G) = .
Proof. For every g ∈ G r  choose (by Zorn’s Lemma) U ≤ G maximal with hgi ∩ U = .
Since o(g) = p it follows that G = hU ∪ {g}i, U < G. Hence g ∈
/ Φ(G).
max
5.9.2. Φ(G) ≤ G,
char
because H < G ⇔ H α < G for all α ∈ Aut G.
max
max
˙ . Then Φ(G) ≤ Φ(U)Φ(V ).
5.9.3. Let U, V ≤ G such that G = U ×V
Proof. The mapping U → G/V , u 7→ V u, is an isomorphism. Hence
\
Φ(G) ≤
H = Φ(U)V,
V ≤H < G
max
or H=G
likewise Φ(G) ≤ Φ(V )U. It follows that Φ(G) ≤ Φ(U)V ∩ UΦ(V ) = Φ(U)Φ(V ), because
every element of G has a unique representation uv with u ∈ U, v ∈ V .
The following remark is trivial:
5.9.4. Let U ≤ H for some H < G. Then G 6= Φ(G)U.
max
We call G weakly noetherian if every proper subgroup of G is contained in a maximal
subgroup of G.
5.9.5. Let G be weakly noetherian, T ⊆ Φ(G). Then we have
(∗)
∀X ⊆ G
(hXi = G ⇒ hX r T i = G)
Proof. Let X ⊆ G such that hX r T i =
6 G. By hypothesis, hX r T i ≤ H for some
H < G. Then X ⊆ (X r T ) ∪ T ⊆ H as T ⊆ Φ(G). Hence hXi ≤ H < G.
max
55
Let p be a prime. A p-group G is called extra-special if G is elementary abelian or Φ(G) = Z(G) = G′
is of order p. From 1.0.2 and 1.8(2) it follows easily that every non-abelian group of order p3 is
extra-special.
56
A maximal subgroup of an abelian group has prime index. But nQ = Q for all n ∈ N. Hence (Q, +)
has no proper subgroup of finite index.
62
The converse of 5.9.5 holds without any hypothesis about G: Let T ⊆ G with the
property (∗) and assume that T 6⊆ H for some maximal subgroup H of G. Putting
X := H ∪ T , we then have G = hXi, hence, by (∗), G = hX r T i ≤ H, a contradiction.
Therefore, if G is weakly noetherian, the subsets of Φ(G) are exactly those subsets
of G which can be dispensed with in any set of generators of G. In particular, this
characterization holds in every finite group.
Furthermore, it should be noted that the assertion of 5.9.5 holds for an arbitrary group
G if T is finite. It suffices to prove this for the case |T | = 1 as the claim for arbitrary
finite T then follows by routine induction. Let X be a set of generators of G and g
the unique element of T . If we assume hX r {g}i =
6 G, we find by Zorn’s Lemma a
subgroup H which is maximal subject to the conditions that X r {g} ⊆ H, g 6∈ H. For
all y ∈ G r H the subgroup hH ∪ {y}i then contains g, hence X, and therefore equals G.
Thus H < G. But then g ∈ Φ(G) ≤ H, a contradiction. For this reason the elements
max
of Φ(G) are also called the non-generators of G.
5.9.6. Let G be weakly noetherian, X ⊆ G. Then hXi = G ⇔ h{Φ(G)x|x ∈ X}i =
G/Φ(G).
Proof. Clearly, the cosets of the elements of a set of generators modulo a normal subgroup generate its factor group. If hΦ(G)x|x ∈ Xi = G/Φ(G), then hΦ(G) ∪ Xi = G,
hence hXi = G by 5.9.5.
5.9.7. Let N E G, N weakly noetherian. Then Φ(N) ≤ Φ(G).
Proof. Let H < G and assume Φ(N) 6≤ H. From 5.9.2 we conclude Φ(N)H = G, hence
max
N = Φ(N)(H ∩ N). Then, by 5.9.4, H ∩ N = N so that Φ(N) ≤ H, a contradiction.
Clearly, direct factors of weakly noetherian groups are weakly noetherian. Therefore we
˙ implies Φ(G) = Φ(U)Φ(V ) if G is weakly
conclude from 5.9.3 and 5.9.7 that G = U ×V
noetherian.
We shall need the following simple remark:
5.9.8. Let A be an abelian normal subgroup of G and S ≤ G such that SA = G. Then
S ∩ A E G.
Proof. S ∩ A E S because A E G, and S ∩ A E A because A is abelian.
5.10 Proposition. Let G be a group.
(1) Let Φ(G) ≤ N E G. If N/Φ(G) is nilpotent, then N/Φ(G) is abelian.
(2) An abelian minimal normal subgroup A of G has a complement in G if and only if
A 6≤ Φ(G).
63
Proof. (1) Without loss of generality we may assume that Φ(G) = , N is nilpotent.
We have to show that N ′ =  and assume that N ′ 6= . Then there exists a maximal
subgroup H of G such that N ′ 6≤ H, hence HN ′ = G. Let D := H ∩ N. Then there
exists a smallest j ∈ N such that DZj (N) = N. Put M := Zj−1(N). Then D ≤ M < N,
[N, M] = [DZj (N), M] ≤ [D, M][Zj (N), M] ≤ M, hence M ⊳ N by 5.0.3, and N/M is
abelian, being isomorphic to a factor group of Zj (N)/Zj−1 (N). It follows that N ′ ≤ M,
implying N = (H ∩ N)N ′ = DN ′ ≤ M, a contradiction.
To prove (2), suppose first that a minimal abelian normal subgroup A has a complement
H in G. If H ≤ S < G, 5.9.8 implies that S ∩ A E G. Furthermore,  ≤ S ∩ A < A.
Hence S ∩ A = , H = S. Thus H < G and A 6≤ Φ(G). Conversely, if A 6≤ Φ(G), there
max
exists a maximal subgroup H of G such that A 6≤ H. It follows that HA = G and, by
5.9.8 and the minimality of A, H ∩ A = .
5.11 Theorem. Let G be a group, N E G, N finite, D := N ∩ Φ(G). The following are
equivalent:
G
(i) N is nilpotent,
(ii) N/D is abelian,
finite
N

D
(iii) N/D is nilpotent.
In particular, if N ≤ Φ(G), then N is nilpotent.
Φ(G)

Proof. (i) implies (iii) by 5.1.6, and (iii) implies (ii) by 5.10(1) as NΦ(G)/Φ(G) ∼
= N/D.
N
G
M
NG (P )
D
P

Suppose (ii) and let P ∈ Syl(N), M := DP . As N/D is abelian and
M/D ∈ Syl(N/D) we have M/D ≤ N/D. It follows that M E G.
char
Since P ∈ Syl(M), 1.12 implies that G = M NG (P ) = D NG (P ),
hence G = Φ(G)NG (D). By 5.9.4, NG (P ) cannot be contained in
any maximal subgroup of G. But |G : NG (P )| ≤ |D| ≤ |N| is finite.
Hence NG (P ) = G, P E G. Now (i) follows from 5.3.
If N ≤ Φ(G), then (ii) and (iii) hold trivially, hence (i).
5.12 Corollary. Let G be a finite group.
(1) (Frattini 1885) Φ(G) is nilpotent.
(2) (Wielandt 1937) G is nilpotent ⇔ G′ ≤ Φ(G).
(3) (Gaschütz 1953) G is nilpotent ⇔ G/Φ(G) is nilpotent.
(4) (Gaschütz 1953) F (G)/Φ(G) = F (G/Φ(G)).
F (G)/Φ(G) is abelian, is the product of all minimal abelian normal subgroups of
G/Φ(G) and has a complement in G/Φ(G).
64
Proof. (1) is clear by the last assertion in 5.11. (2) follows from the equivalence of (i)
and (ii), (3) from the equivalence of (i) and (iii) in 5.11, putting N := G.
(4) By (1) we have Φ(G) ≤ F (G). Moreover, 5.1.6 implies that F (G)/Φ(G) ≤ F (G/Φ(G)).
The converse inclusion holds because (iii) implies (i) in 5.11, where we let N be the
normal subgroup of G such that Φ(G) ≤ N, N/Φ(G) = F (G/Φ(G)). By 5.10(1),
F (G)/Φ(G) is abelian. For the remaining assertions we may assume w. l. o. g. that
Φ(G) = . Let N be a maximal product of minimal abelian normal subgroups of G
such that N has a complement K in G. Clearly, N ≤ F (G). Assume F (G) ∩ K 6= .
By 5.9.8, F (G) ∩ K E G so that there exists a minimal normal subgroup A of G in
F (G) ∩ K. By 5.10(2), A has a complement in G, hence also in K. A complement of
A in K, however, is a complement of NA in G, in contradiction to the choice of N. It
follows that F (G) ∩ K = , N = F (G). Thus F (G) is the product of all minimal abelian
normal subgroups of G.
5.13 Theorem (Roquette 1964). Let G be a group, π a set of primes, N E G, N finite
and N [π] = N. Suppose H ≤ G such that H ∩ N ∈ Hallπ (N). If H ∩ N ≤ Φ(H), then
H ∩ N = .
G
H
N
Φ(H)
M

Proof. Put M := H ∩ N. Then the hypotheses of 4.8 (with M ∗ = M ′ )
are satisfied. As M ≤ H ≤ NG (M), there exists a subgroup S of H
such that SM = H, S ∩ M = M ′ . Since |H : S| is bounded by |M|,
hence finite, 5.9.4 implies S = H. We thus obtain M ′ = H ∩ M =
M ≤ Φ(H). By 5.11, M is nilpotent so that 5.1.4 implies M = .
Special case:57 Let G be a finite group, p a prime, P ∈ Sylp (G) and P ∩O p (G) ≤ Φ(P ).
Then P ∩ O p (G) = .
5.14 Corollary. Let H be a finite group. Then every prime divisor of |CAut H (H/Φ(H))|
is a divisor of |Φ(H)|. 58 If H is a p-group, then CAut H (H/Φ(H)) is a p-group.
Proof. Assume that p ∤ |Φ(H)| for some prime divisor p of |CAut H (H/Φ(H))|. By 1.9(4),
CAut H (H/Φ(H)) has a subgroup A of order p. Let G := H ⋊ A, N := [H, A]A.
id
G
H
N
Φ(H)
A
[H, A]
p
Then [H, A] ≤ Φ(H)
= H ∩ N ∈ Hall(N)
≤ H, hence N E G, [H,[pA]
′]
as p = |N : [H, A]| ∤ |[H, A]|. We have A ≤ N , hence
′
′
[H, A] ≤ [H, N [p ] ] ≤ H ∩ N [p ] ≤ H ∩ N ≤ [H, A]
′

so that N [p ] = N. Now 5.13 implies that H ∩ N = , i. e., [H, A] = ,
a contradiction since H cannot be centralized by a nontrivial subgroup of Aut H.
57
58
Clearly, this also follows easily from 4.9(1).
P. Hall (1933) proved the following stronger result: If H has a system of d generators, then
|CAut H (H/Φ(H))| is a divisor of |Φ(H)|d (see [H], III, 3.17 for a proof).
65
5.15 Proposition. Let G be a group, A ≤ Aut G. Suppose that A and [G, A] are finite
and of coprime orders.
(1) G = CG (A)[G, A].
˙
(2) If G is abelian, then G = CG (A)×[G,
A].
In particular, (1), (2) hold if G is a p-group and A a p′ -subgroup of Aut G. If then
[G, A] ⊆ Φ(G), it follows that A = .
Proof. (1) follows from 3.14, putting H := [G, A].
Q
Q
(2) Let ϕ : G → G, g 7→ α∈A g α . Then ϕ ∈ End G and [g, β]ϕ = α∈A (g −1 )α g βα = 1G
for all g ∈ G, β ∈ A. For x ∈ CG (A) ∩ [G, A] it follows that x|A| = xϕ = 1G which under
our hypothesis implies x = 1G .
If G is a p-group and A ≤ Aut G such that [G, A] ⊆ Φ(G), then A ≤ CAut G (G/Φ(G)),
hence A is a p-group by 5.14.
We illustrate 5.15(2) by a typical application (cf. p. 51):
5.16 Corollary. Let G be a finite group, p a prime. Suppose that a Sylow p-subgroup
P of G is abelian. Then CP (NG (P )) is a complement of G[p] in G.
Proof. By 4.4, P ∩ G′ = [P, NG (P )] = [P, A] where A is the image of NG (P ) in Aut P
with respect to the action of NG (P ) on P by conjugation. We have NG (P ) ≥ CG (P ) ≥
P ∈ Sylp (G), hence A, being isomorphic to NG (P )/CG (P ), is a p′ -group. By 5.15(2),
˙
P = CP (A)×[P,
A]. It follows that G = P G[p] = CP (A)G[p] and CP (A) ∩ G[p] ≤
CP (A) ∩ [P, A] = . As CP (A) = CP (NG (P )), the proof is complete.
In p-groups, the Frattini subgroup may be characterized in a more specific way than
in arbitrary finite groups as will become clear in a moment. For any finite group and
any prime p there is a smallest normal subgroup the quotient of which is an elementary
abelian p-group: Any such subgroup must contain G′ and the set Gp of all g p (g ∈ G).
The latter is generally not a subgroup, but in an abelian group it is. It follows that
G′ Gp is a subgroup of G, and clearly normal with an elementary abelian factor group.
Hence we have
5.16.1. Let G be a finite group, p a prime, N ≤ G. Then N E G and G/N is an
elementary abelian p-group if and only if G′ Gp ≤ N. Furthermore, Φ(G) ≤ G′ Gp ,
the last assertion being a consequence of 5.9.1.
Every abelian group M is a Z-module with respect to the natural action
M → M
ϕ : Z → End M, z 7→
g 7→ g z
If M 6=  and xp = 1M for all m ∈ M, it follows that ker ϕ = pZ, hence ϕ induces an
action of the prime field Fp := Z/pZ on M so that M becomes a vector space over Fp .
The Fp -subspaces of M are just the subgroups of M. If M is finite and |M| = pd where
d ∈ N0 , then d = dimFp M.
66
5.17 Burnside’s basis theorem. Let p be a prime and G a p-group.
(1) Φ(G) = G′ Gp .
(2) Any two minimal sets of generators of G are equipotent. Their cardinality is the
number d such that |G/Φ(G)| = pd .
Proof. (1) Φ(G) ≤ G′ Gp by 5.16.1. By 5.2(2), every maximal subgroup of G is normal,
hence of index p as the factor group has no nontrivial subgroup. Thus Gp ⊆ Φ(G). As
G′ ≤ Φ(G) by 5.12(2), the claim follows.
(2) Let X be a minimal set of generators of G. The mapping
X → G/Φ(G), x 7→ Φ(G)x,
is injective because y ∈ Φ(G)x, x, y ∈ X, x 6= y, implies h(X r {y}) ∪ Φ(G)i = G,
hence hX r {y}i = G by 5.9.5, a contradiction. Furthermore, the set of generators
{Φ(G)x|x ∈ G} of G/Φ(G) is minimal, by 5.9.6. Thus any minimal set of generators of
G maps bijectively onto a basis of the Fp -vector space G/Φ(G) which is of dimension d.
The claim follows.
67
A Appendix and Outlook
Recall that the class of all soluble groups is closed with respect to extensions: If N is
a soluble normal subgroup of a group H such that H/N is soluble, then H is soluble.
In particular, a product of finitely many soluble normal subgroups is a soluble normal
subgroup59 . As in the case of the class of nilpotent groups (cf. 5.7), this observation
gives rise to the following definition:
A.1 Definition. Let G be a group. Then
Y
Sol(G) :=
N
NEG
Nsoluble
is called the soluble radical of G. If G is finite, then Sol(G) is soluble and Sol(G/Sol(G))
=  because the class of all soluble groups is closed with respect to extensions. Thus
there is a natural general programme to study finite groups, consisting of the following
three parts:
• Describe all soluble groups
• Describe all groups with trivial soluble radical
• Describe all extensions of soluble groups by groups with trivial soluble radical.
We will add a number of observations about each of these three points. As a result we
will see which topics constitute key chapters in the pursuit of the above programme.
This may serve as a help for an orientation about current research efforts in finite group
theory.
The socle of G is defined by
Soc(G) :=
Y
M.
M EG
min
If G is finite and soluble, every chief factor of G, being a minimal normal subgroup of a
quotient of G, is elementary abelian. Furthermore, 5.12(4) implies the following remark:
A.1.1. If G is a finite soluble group, then Soc(G/Φ(G)) = F (G)/Φ(G).
59
In other words, the class of soluble groups is closed with respect to finite normal products. Note
that this remark is almost trivial in contrast with the analogous assertion for the class of nilpotent
groups (see 5.6).
68
From 2.12 we know that in the finite case a minimal normal subgroup M has the form
S × · · · × S for some simple group S and some n ∈ N.
|
{z
}
n
(Aut S)n
Aut M
If S is nonabelian, then Aut M ∼
= Sn ≀ Aut S, by 3.2(2). The normal
id
(In S)n

Sn
subgroup In S of Aut S is isomorphic to S. Hence Aut M is an extension of a normal subgroup ∼
= M and Sn ≀ Out S where Out S :=
id
Aut S/In S, called the outer automorphism group of S.60 For example,
let S = An for some n ≥ 5. By 1.3, S is simple. As An E Sn , the action of Sn on
An given by conjugation is a monomorphism of Sn into Aut An . It can be shown that
this monomorphism is an isomorphism unless n = 6. Thus, for n 6= 6, |Out An | = 2.
Furthermore, |Out A6 | = 4 (see, for example, [H], II, 5.5).
A.2 Proposition. Let G be a finite group such that Sol(G) = . Let M1 , . . . , Mk be the
˙ · · · ×M
˙ k as
(mutually distinct) minimal normal subgroups of G. (Then Soc(G) = M1 ×
the Mi are non-abelian.) Let
f : G → (Aut M1 ) × · · · × (Aut Mk ), g 7→ (ḡ|M1 , . . . , ḡ|Mk ),
where ḡ denotes the conjugation by g. Then f is a monomorphism.
Proof. Each Mi is invariant under conjugation in G, hence f is a homomorphism and
ker f = CG (Soc(G)). Assume that ker f 6= . Then ker f contains a minimal normal
subgroup of G, w. l. o. g. M1 ≤ ker f . Thus [M1 , M1 ] ≤ [Soc(G), CG (Soc(G))] = , i. e.,
M1 is abelian, a contradiction. Hence ker f = , f is injective.
In the sense of A.2, the structure of G is controlled by the socle of G if G is finite and
Sol(G) = : The groups G with a given socle (more precisely, with prescribed minimal
normal subgroups M) may be found as subgroups of the direct product of the groups
Aut M the structure of which was considered above, if G is finite and Sol(G) = . Of
interest are the subgroups which contain In M for every M and act transitively on the
set of simple direct factors S of M. The main unknown in this approach is the structure
of Aut S. Roughly spoken, the description of all finite groups G with Sol(G) = 
along the above lines depends on the knowledge of all finite simple groups S and their
automorphism groups.
We have already seen (cf. 5.8.1 and the subsequent comments) that the structure of a
finite soluble group G is controlled in a similar way by F (G): Given any finite nilpotent
group N, the finite soluble groups G such that F (G) = N have the property that
G/Z(N) may be found, up to isomorphism, as subgroups of the direct product of the
groups Aut P containing In P as their largest normal p-subgroup, where P ranges over
60
The famous Schreier conjecture claims that the outer automorphism group of a finite simple non-cyclic
group is always soluble. Having been undecided for decades, this has been proved – like a number
of other long-standing conjectures in group theory – on the basis of the classification of all finite
simple groups.
69
Syl (N). In this sense, the description of all finite soluble groups allows a reduction to
the study of all p-groups and their automorphism groups, plus the extension problem of
an abelian normal subgroup V (∼
= G/Z(N)).
= Z(N)) by a finite soluble group U (∼
The general extension problem asks for a description of all groups which have a given
normal subgroup V with a likewise given quotient U. The best understood extensions are
the splitting ones, given by the semidirect products of V and U, introduced and studied
in Chap. 3. Therefore, the split extensions of V by U may be viewed as the trivial
cases of the general extension problem. They depend on the possible group actions of
U on V and nothing more. Conditions on the groups U, V which guarantee that every
extension of V by U must necessarily split constitute therefore a most satisfactory and
useful contribution to the extension problem, solving it completely in those cases. The
famous theorems 3.9 and 3.13 are results of this kind. But without restrictive hypotheses
on V and U there will, in general, exist non-split extensions which are not accessible
by the methods treated in Chap. 3. An interesting connection between the field of all
extensions (of two given groups) and that of split extensions is, however, established by
means of the wreath product as has been proved in 3.7.
A major special case, for many problems the real core, is that of an abelian normal
subgroup. In particular, we have seen this to be true with respect to the aspect of a
finite soluble group as an extension of the centre of its Fitting subgroup. Moreover,
the problem of extending a soluble normal subgroup V may be viewed as an iteration
of extending abelian normal subgroups (in the first step extending V /V ′ by the desired
quotient, then V ′ /V ′′ , by the quotient obtained in the foregoing extension step, etc.).
Thus we also have a reduction of the third point in the list on p. 68 to the case of an
abelian normal subgroup.
Let G be an extension of an abelian group V by a group U. Then V , being abelian,
is contained in the kernel of the action of G on V . Hence each such extension G is
associated with some action of U on V . As in the case of the semidirect product we
may thus start from a given action of U on V and restrict ourselves to the study of
possible extensions G which induce that particular group action of U on V . Thus we
view V as an U-module, fixing a certain action of U on the abelian group V right from
the beginning. A tiny example is given by V ∼
=U ∼
= Cp for a prime p, where the only
possible action is the trivial one. Clearly, each extension of V by U is of order p2 and
abelian (cf. also 1.8.1). In this extremely simple special case of 2.11 there are exactly
two distinct isomorphism types: the cyclic one and the elementary abelian one. But
even in this little example we see already that there is not only the split extension (the
elementary abelian group) but also a non-split one (the cyclic group). Recall that this
cannot happen if gcd(|U|, |V |) = 1, by 3.13. But as soon as this hypothesis of coprime
orders is not satisfied we have to expect a nontrivial situation, i. e., more than just the
splitting extension of V by U. A simple example with nontrivial group action is given by
V ∼
= C2 , where the nontrivial element of U induces the automorphism v 7→ v −1
= C4 , U ∼
of V . The extensions with respect to this group action are represented by the dihedral
group D8 (the split extension) and by the quaternion group Q8 (a non-split extension).
70
An important tool for investigations about modules of a group is cohomology theory. The
so-called n-th cohomology group with respect to an action f of a group U on an abelian
n
group V is a certain factor group of a subgroup of (V U , +̇) (see 2.3). For example, if
n = 1, this subgroup is given by the crossed homomorphisms of U into V (see 3.3). One
readily verifies that for every v ∈ V the mapping wv : U → V, u 7→ v −1 v uf , is a crossed
homomorphism. The factor group of the group of all crossed homomorphisms by the
subgroup consisting of the mappings wv (v ∈ V ) is the first cohomology group. The
second cohomology group (n = 2) arises in a more complicated but principally similar
way. The important link with the extension problem is the fact that each of its elements
allows an interpretation as a group extension of V by U (respecting the given action f )
and that all of these extensions are obtained by means of (at least) one element of the
second cohomology group. Its neutral element corresponds to the semidirect product of
V by U with respect to f . Hence the above remark that the latter may be viewed as
the trivial case of an extension of V by U has a concrete structural background.
To study extensions therefore means to determine the second cohomology group. This
may look like an isolated detail at the first glance, but in fact there are strong connections
between the cohomology groups for different n ∈ N so that the whole theory plays a role
even if in some context (like here) only a particular cohomology group is of interest.
Summarizing, we may consider the following topics as important chapters in finite group
theory in regard of the programme sketched on p. 68:
• p-groups and their automorphisms groups,
• finite simple groups and their automorphism groups,
• cohomology theory (in particular: extension theory).
The remarks in this appendix result in a modified version of what may be viewed as
Hans Fitting’s legacy, his programmative paper [F]. Since the days of Fitting, enormous
progress has been made with respect to the second point. If we may trust in what has
been claimed by experts of the area of simple groups for many years, the classification
is complete. While [W] is a detailed description of all finite simple groups, hence is a
resultative account after the efforts of several decades of hardest work, the project of a
complete presentation of a proof of the classification itself, however, is still unfinished
although apparently well on its way. On the other hand, the first and the third of the
above-mentioned points are still far from a satisfactory understanding although much
energy has already been put into those areas too. Thus group theory is and will probably
remain for a long time a research area with deep secrets to be discovered. Moreover, it
should be clear that Fitting’s programme and also a modernized version of it is only one
possible kind of orientation in this fascinating universe. From a certain point of view it
may help to understand main streams in group theory. But group theory is certainly
a more complex field. The reader should bear in mind that the importance of groups
lies, to a considerable extent, in their occurrence in other mathematical theories. While
it is true that group theory has become a highly elaborate and specialized building of
71
notions and arguments, it is likewise true that those connections have had an important
influence on group theory. In this course we have presented purely group-theoretic
methods and concepts which allow to penetrate into many directions of modern research.
Our presentation should not be interpreted as a narrow-minded limitation to a so-called
purely group-theoretic position. But for a fruitful combination with other areas inside
or outside Algebra, a thorough knowledge of the rich spectrum of purely group-theoretic
concepts is, in our opinion, the best preparation.
72
Bibliography
[B]
Baer, Reinhold, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133
(1957), 256–270.
[FT]
Feit, Walter and Thompson, John G., Solvability of groups of odd order, Pacific
J. Math. 13 (1963), 775–1029.
[F]
Fitting, Hans, Beiträge zur Theorie der endlichen Gruppen, Jber. Dtsch. Math.Ver. 48 (1938), 77–141.
[G]
Gorenstein, Daniel, Finite Groups, Harper & Row, 1968
[GI]
Gagola, Jr., Stephen M. and Isaacs, I. M., Transfer and Tate’s theorem.
Arch. Math. 91 (2008), 300-306.
[H]
Huppert, Bertram, Endliche Gruppen I, Springer, 1967
[I]
Isaacs, I. Martin, Finite Group Theory, Grad. Stud. in Math. 92, AMS, 2008
[KS]
Kurzweil, Hans and Stellmacher, Bernd, The Theory of Groups, Springer, 2004
[L]
Lam, T. Y., A First Course in Noncommutative Rings, Springer, 20012
[R]
Robinson, Derek J. S., A Course in the Theory of Groups, Springer, 19962
[W]
Wilson, Robert A., The Finite Simple Groups, Springer, 2009
73
Index
crossed homomorphism, 39
cycle, 6
cycle decomposition, 7
Ω-decomposition, 26
action, 12
ascending chain condition, 29
automorphism, 5
automorphism
inner, 5
automorphism group, 5
automorphism group
inner, 6
derived series, 45
descending chain condition, 29
dihedral group, 35
direct Ω-decomposition, 24
direct Ω-factor, 24
directly Ω-indecomposable, 24
disjoint cycles, 7
bipartite Ω-decomposition, 24
elementary abelian, 31
Engel element, 60
equivalent sets under an action, 15
extra-special, 62
central chain, 55
central series
ascending, 55
descending, 55
centralized group action, 17
centralizer, 14
centre, 5
chain
ascending, 55
descending, 55
characteristic, 23
chief factor, 32
chief series, 32
commutator subgroup, 23
compatible, 15
complement, Ω-, 24
complete wreath product, 37
composition factor, Ω-, 32
composition series, Ω-, 32
conjugacy class, 6
conjugation, 5
constrained, 61
control of transfer, 52
core, 14
factor, Ω-, 24
Fitting subgroup, 59
focal subgroup, 50
Frattini subgroup, 62
fully invariant, 23
group
π, 18
-p, 18
-p′, 18
alternating, 8
simple, 9
group action, 12
group action
elementwise fixed-point-free, 13
faithful, 12
transitive, 13
group with operators, 12
Hall subgroup, 45
74
semidirect decomposition, 25
semidirect product, 34
sign homomorphism, 8
similar sets under an action, 15
similar subnormal series, 32
simple, Ω-, 24
socle, 68
space
K-, 23
split extension, 35
stabilizer, 9
step number, 45
subgroup
maximal, 9
subnormal, 31
subnormal series, 31
Sylow subgroup, 20
holomorph, 36
homomorphism, Ω-, 24
hypercentre, 55
inverse left multiplication, 13
left Engel element, 60
left-normed commutator, 58
length of a cycle, 7
length of a subgroup series, 31
module, 23
monomial matrix, 38
nilpotency class, 56
nilpotent group, 55
nilpotent residual, 57
non-generators, 63
normal series, 31
normal set of operators, 26
normal subset, 6
normalized group action, 16
normalizer, 14
transfer, 48
transposition, 8
transvection, 11
type of an abelian p-group, 30
operating system, 12
operator, 12
orbit, 13
orbit
π, 6
nontrivial, 6
outer outomorphism group, 69
uniquely n-divisible, 44
weakly noetherian, 62
wreath product, 36
partition of a positive integer, 30
permutation
even, 8
permutation matrix, 38
regular wreath product, 37
replacement automorphism, 26
representation
of a K-space, 23
of an algebra, 24
retracting Ω-factor, 27
right Engel element, 60
right multiplication, 13
Schreier conjecture, 69
75