ON THE STRUCTURE OF GROUPS OF FINITE ORDER
RICHARD BRAUER
The theory of groups of finite order has been rather in a state of stagnation
in recent years. This has certainly not been due to a lack of unsolved problems.
As in the theory of numbers, it is easier to ask questions in the theory of groups
than to answer them. If I present here some investigations on groups of finite
order, it is with the hope of raising new interest in this field. The results which
I am going to give will be incomplete and new unanswered questions will be
added to the old ones. There is, however, hope that further progress will be
possible along the lines suggested.
Let me start with the following question. If © is a group of order g which
is not cyclic of prime order, what can be said about the existence of proper subgroups § of relatively large order hi It seems probable that the following conjecture is true. There always exist proper subgroups § of an order h >tyg. For
instance, if © is soluble, it is easy to see that there exist proper subgroups §
even with h ^ y g. There is a famous conjecture to the effect that all groups of
odd order are soluble. Thus, if we accept this old conjecture, we have to consider
only groups of even order. It' is somewhat surprising that for groups of even
order our conjecture can be proved. In other words, we have the following
theorem
Theorem. If © is a group of even order g > 2, there exists a proper subgroup
§ of an order h > <s*/g.
The proof is quite elementary and I will sketch it, since it forms a preparation for some of the later developments. It will be convenient to work with the
group algebra r of © over the field of rational numbers. Thisfe an associative
algebra consisting of all formal expressions
y = Haaa
a
where a ranges over the elements of © and where the aa are rational numbers.
The operations in F are defined in the obvious manner.
By an involution of ©, we shall mean an element / of © of order 2. Letali
denote the set of all involutions in ©. Since g is even, SUI is not empty. Set
M=I,J
leM
taken as an element of r. Then M2 e T and we have a formula
209
M 2 = S aaa.
(1)
a
It is clear that aa here is the number of ordered pairs (X, Y) such that
(2)
XY = G,
(X,Yem).
It is also clear that aa will depend only on the class of conjugate elements of ©
to which a belongs. If then the classes of conjugate elements of © are denoted
by S 0 , ®lf . . ., S*;-! and if i£< e jTis the sum of the elements of ffiif the formula
(1) can be rewritten in the form
fc-1
M2=HaiKi
(1*)
where we set <^ = aa for a e ®$.
If (2) holds, it follows at once that X~XGX = a"1. We shall say that an
element a is real in ®, if a and o*"1 are conjugate in ®. Thus, if a is non-real in
®, (2) is impossible and at = 0 for non-real classes ®<B Moreover, if 5R(or) is the
normahzer of a in ©, (i.e. the subgroup of © consisting of the elements commuting with a), and if n(a) is the order of 31(a), there exist at most n(a) elements X
with X~xaX = or"1. Thus, if ai is an element offfi^we have ai ^ w(crz-). If at is
an involution, one sees easily that this inequality can be replaced by
&% ^ n[<fi) — 2. Actually, it will be clear that much more accurate statements
concerning the values of ai can be given, but this is not important for the
moment.
Let us now compare the number of group elements appearing on both sides
of (1*). If m denotes the total number of involutions in ®, the left side of (1*)
is the sum of m2 elements of ®. On the right, we have a sum of yEiaig/n(ai) elements of ®, since ffi^ consists of gfnfa) elements. Hence
m2 = S
agjnfai).
Let S 0 denote the class containing the identity 1. Clearly, a0 = m. Let
$èv . . ., $r designate the classes consisting of involutions and let S r + 1 , . • ., Sj
designate the other classes consisting of real elements of ®. If the estimates
for the at are used, we find
m2^m
r
i
+ Jl (n(at) — 2)g/»(cri) + S »(flOg/nfo).
i=l
j=r+l
Each of the m involutions appears in one of the classes &v ffia, . . ., S r , and hence
r
m = 2 gln(öi).
i=i
Thus,
i
(3)
m2 < S n(Gi)gln(Oi) = lg.
2=1
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On the other hand, the total number of elements in $èv S2> • • •» ®i *s * ess than g,
that is,
i
(4)
2 g/«(er,) < g.
Let h be the maximal value of n(ai) appearing in (4). Then g/h 5g gjn(ai) and
(4) yields
lg/h < g.
If this is used in (3), we have m2 < gh. On the other hand, m ^ g/^(c1) ^ g/A.
On combining these inequalities, we find g < hs. If h = n(<r3-) 7^ g, then 91 (a*,-)
is a proper subgroup § of an order h > -y/g. It remains to discuss the case that
the maximal n(öj) in (4) is equal to g. In this case, ®3- consists of one element
only. For r + 1 ^ i ^ I the class ^ contains the two distinct elements oi and
af1. Hence we must have 1 fg 7 ^ r, and then o^ is an invariant involution.
Now, induction can be used. If G/{o^} has even order, the theorem holds for
this quotient group and we see at once that it then holds for ©. If ©/{cr,} has
odd order g', then the order g of © is twice an odd number g'. It is well known 1 )
that then © has a subgroup or order g/2. Since g/2 > ^/g for g ^ 4, the theorem
holds again and this finishes the proof. Incidentally, it is possible to improve
the theorem somewhat, but it does not seem to be worth while to go into this
here. Instead I rather mention a related theorem which is proved by a similar
method.
Theorem. Let m denote the number of involutions of the group © of even order
g and set n = gjm. There exists a subgroup § ^ © of © such that the index
t =(©:<£)) is either 2 or t satisfies the inequality t ^ n(n + 2)/2.
This result has some interesting consequences. If the group © has a subgroup § ^ © of index t, then © possesses a transitive permutation representation on t symbols. The kernel £ of this permutation representation is a normal
subgroup of © such that ©/S is isomorphic with a transitive subgroup of the
symmetric group ©$ on t letters. This implies that
K
(© : S) ^ t\ ^ Max (2, [n(n + 2)/2]l).
In particular, if © is simple, we must have ß = {1} and hence
g ^ [ » ( » + 2)/2]l.
If / is any involution in ©, then g/n(J) fg m = g/n and hence n
g ^ (n(J)(n{J) + 2)/2)l This yields the result 2 ).
x
S^n(J),
) See, for instance, W. Burnside, The Theory of Groups of Finite Order, Cambridge
1911, p. 327, Theorem II.
2
) A result of this kind with a different bound was first obtained by K. A. Fowler and
the author by a somewhat different method.
211
Theorem. There exist only a finite number of simple groups © which contain
an involution J such that the normalizer 3l(J) of J is isomorphic to a given group.
It is not known whether a similar result holds, if, instead of an involution
J, we take an element of a given odd prime order p.
The last theorem suggests the following problem: Given a group 31 containing an involution / in its center. What are the groups © containing 31 as a
subgroup such that 31 is the normalizer of / in ®? More generally, the case can
be considered that a number of groups 31*, 31*, .. -, 31* are given such that each
3li contains an invariant involution Jim We then cari study the groups ® containing subgroups $ft*, 31*, • . ., 31* such that each 31* is isomorphic to 3li and
that 31* is the normalizer of the element / * corresponding to Jt. We may
further require that / * , / * , • • • , / * represent all the distinct classes of ® which
contain involutions. Usually, it will be convenient to add further assumptions
in order to characterize groups uniquely.
In this generality, the problem is far too difficult to handle. There are,
however, some general methods which give at least some information and which
we shall discuss next.
The first of these is closely related with what we have done above. If we use
the same notation, the elements Klf K2, . . .,Kk form a basis of the center A
of the group algebra 71. We therefore have formulas
KaKß = S caßyKy
with rational coefficients caßy. Actually, it is easy to see that the caßy are nonnegative integers. The caßy can be expressed in terms of the irreducible characters of ® and, as a matter of fact, it is well known that a knowledge of the caßy
is equivalent with a knowledge of the characters.
Suppose now that g is even. The element M in (1 *) is the sum oiKlt K2, . . .
Kr. Hence theflÄ-in (1*) are sums of certain of the caßy. Since we can study the
ai directly and since we can express the caßy in terms of the characters, we obtain
new relations for the characters.
The second idea is of a more fundamental nature. It applies to groups of
even or odd order and it can be used for other questions too. Let %x, %2, . . ., %h
designate the irreducible characters of ®. Let p be a fixed prime number. In
our application, we take p = 2, but this is not important now. The characters
Xi are distributed into disjoint sets, called the p-blocks of ®. Here, %a and %ß
belong to the same block, if
for all a e ® where p is a fixed prime ideal divisor of p in the field of the g-th
roots of unity. This is a technical definition of the blocks. There is a better one
212
which belongs to the theory of arithmetic of algebras, but it would take too
long to discuss this here 3 ).
Suppose then we have a block B of characters, say, consisting of the
characters %x, %%, . . ., %s. We form the highest power pd which divides one of
the numbers g/Dgxif 1 ^ i ^ s. Then d is called the defect of B. The first basic
result is, that for given p and d and all groups of finite order, there exist only
a finite number of "types" of ^-blocks of defect d. For each type, the number s
of characters of the block is determined. Further, for each type, one can give
explicitly all relations
which hold simultaneously for all elements a of ® of order prime to p. If pa is
the highest power dividing g, one may assume that, for each type, the values
of the degrees T>g%i (mod pa-d+lyj are known.
In particular, for given p and d, there must exist an upper bound for the
number s of characters in the block. It is probable that s ^ pd. However, this
can only be proved for d ^ 2. The best known result, proved recently by W.
Feit and myself, is that s < ^p2d(îov d > 1). Actually, for d ^ 1, all possible
types can be enumerated. For larger d, the number of types may be uncomfortably large. It would be very important for our work if the theory of blocks could
be refined.
There is another result which I have to mention. If all subgroups © of
order pd of a ^-Sylow group $ of © and their normalizers 9?(®), in © are known,
the number of ^-blocks of defect d can be determined and information about
the possible types obtained. For d = 0, we have © = 3l{^) and nothing can be
learned about the number of ^-blocks of defect d = 0. For d > 0, a real reduction can be achieved.
These results can be applied, if we have information about © as outlined in
the program above. The prime p is to be taken as 2. The method will give us a
wealth of information about the characters of © and this again can be applied
to study the group © further.
Let me now discuss some special cases. I first consider a relatively simple
case. Consider a group © of an order g = 4g' where g' is odd. In order to have
more definite results, we shall assume that no element outside a Sylow
subgroup © of order 4 commutes with all the elements of ©. Further, we
shall assume that © does not have a normal subgroup of index 2. The
3
) Some of the results on blocks mentioned here have been stated without proof in
the following papers: R. Brauer, On the arithmetic in a group ring, Proc. Nat. Acad. Sci.
U.S.A. 30, 109—114 (1944) and On blocks of characters of groups of finite order I, II,
Proc. Nat. Acad. Sci. U.S.A. 32, pp. 182—186, 215—219 (1946).
213
normalizer 31{J) of an involution / has an order n(J) of the form n(J) = 4v
with an odd v. If our methods are applied here, the result is obtained that
the order g can be expressed by two integral rational parameters flt f2 and
a sign ô = i 1. We have
^
(/i+ !)(/>+ * ) ( / . - ! )
where we set / 3 = 1 + fx + ôf2. We must also have fx = 1 (mod 4), / 2 = ô
(mod 4). Actually, 1, fv d2, / 3 are degrees of irreducible representations and all
other such degrees are divisible by 4. As a strange consequence of our result
we see that g will he very close to 32ü 3 . In particular, if © is to be simple, then
/i> /2> fs will increase with g and g is asymptotically equal to 32v3.
There do exist infinitely many simple groups of the type in question. They
are groups PSL(2, q)*) i.e. groups of unimodular projectivities of the projective line of a finite geometry belonging to a field with q elements. The prime
power q > 3 has to satisfy the conditions q = 3 or 5 (mod 8) in order that ©
meets the requirements. Also, q may be taken equal to 4. In the latter case, the
icosahedral group is obtained which is the only group possible for v = 1.
Probably, the groups mentioned here are the only simple groups of the
type in question and, possibly, the only simple groups of an order g = 4g' with
an odd gf. However, this question has to be left open.
In any case, we come close here to a characterization of some of the simple
groups PSL(2, q). Actually, characterizations of all these groups can be given
within the framework of our program. For instance, we have the following
theorem first conjectured by M. Suzuki
Theorem. Suppose that © is a group of even order g such that if two cyclic
subgroups 9Ï and SB of even order have an intersection different from {1}, then %
and S3 both are subgroups of a cyclic subgroup of ®. / / © does not have a normal
subgroup of index 2, then © is a group PSL(2, q) (with q > 2) 6 ).
Actually, under the assumptions of this theorem the characters of © can
be determined completely. In particular, this yields the order g of ® and now a
characterization of the groups in question given by H. Zassenhaus 6) can be
used. There are various extensions and generalizations.
There is some hope that, for the other known simple groups, there exist
similar characterizations. However, the continuation of this work becomes
g
= 32*,*
4
) The notation here is t h a t used in B. L. van der Waerden, Gruppen von linearen
Transformationen, Ergebnisse der Math., vol. 4 (1935).
6
) This theorem was proved more or less independently by M. Suzuki and the author
and b y G. E. Wall.
6
) H. Zassenhaus, Abh. Math. Sem. Univ. Hamburg, 11, 17—40 (1936).
214
more and more difficult and, for this reason, I have been able to complete the
work only in the case of the groups PSL (3, q), the group of unimodular collineations of a projective plane under special assumptions for q. The result here
is as follows
Theorem. Suppose that © is a group of finite order which satisfies the following
conditions
(I) © contains an involution J whose normalizer 3l(J) is isomorphic to
GL (2, q), (the full linear group of degree 2 over a Galois field with q elements).
(II) If C is an element ^ 1 of the center (£ of 3l(J), the normalizer 31(C) of
C in © is 3l(J) (and not larger than 31 (/)).
(III) © is its own commutator subgroup.
If q = — 1 (mod 4), q^k 1 (mod 3), and q ^ 3 then © is isomorphic to
PSL (3, q). If q — 3, we haven the additional case that © can be the simple Mathieu
group of order 7912.
The latter group appears here as a kind of distorted version of the group
PSL (3, 3) of order 5616.
Since the group GL(2, q) can be characterized by means of the preceding
work, this theorem gives a characterization of the group PSL (3, q) for the
values of q in question. There seems to be little doubt that similar characterizations exist for the other values of q too. Some preliminary work indicates that
the hyperorthogonal groups in three dimensions can be treated in a similar
fashion. As was already stated, there is some hope that similar characterizations
exist for the other simple groups though it is not clear whether these results
are accessible to our methods.
I should like now to discuss briefly the ideas of the proof of the last theorem.
If © is actually equal to the group of collineations of a projective plane n, every
involution / o f © leaves fixed a point P of n and all points of a line I not through
P. Thus, to every / there corresponds a pair (P, I) of a point and a line. Instead of building up geometry using points as the fundamental elements, we
can use such pairs (P, I) as the fundamental concept. The various incidence
relations can then be described in terms of the involutions / corresponding to
the pair (P, I). The axioms take the form of group theoretical statements. The
projective plane is thus replaced by the set 3ft of all involutions Je®. The projectivities are given by the transformations
Ta:J+o-iJo
(Jem)
of 3R onto itself where a is a fixed element of ©. Of course, these Ta form the
full projective group PGL(3, q) which for q =£ 1 (mod 3) coincides with the
special projective group.
Suppose now that ® is an arbitrary group which satisfies the conditions
215
of the theorem and consider the set äft of involutions. If we can show that the
axioms of projective geometry hold (in terms of involutions), then3ft becomes
a projective plane. If we can show further that the projectivities are given by
the transformation Ta, it follows that © is isomorphic to PSL (3, q) and we are
finished.
The real difficulty then is to prove certain properties of the set 3ft of involutions which are in no way evident from the given assumptions. As a matter of
fact, we have to use a long detour in order to obtain them. The methods described above suffice in our present case again to determine the characters of ©
and, in particular, the order g. Now the knowledge of the characters provides us
with the necessary information about the involutions and the proof can be
finished.
The problem treated here of characterizing special simple groups is of
interest in connection with the important unsolved problem of determining all
simple groups of finite order. In order to be able to recognize the known simple
groups, we need workable characterizations of these groups. Though we are
certainly very far from a solution of the general problem of the simple groups
of finite order, at least some kind of plan seems to evolve according to which the
problem might be attacked. One hope would be that the only non-cyclic simple
groups are the alternating groups, the finite analogues of the simple Lie groups
and some distorted versions of a few of the latter groups. It would be necessary
to see where these groups fit into our program and to show that no other cases
can arise. It is not possible to say whether this plan will be workable, since
some of the most important links are missing.
As a matter of fact, it is necessary to mention in this connection again the
old conjecture that all groups of odd order are soluble. Of course, this would
now obtain an added significance. No progress has been made on this question
since it was first formulated.
Under these circumstances, it is perhaps of interest to state an equivalent
conjecture. This is based on the following remark which can be proved rather
easily. If © is a group, if ®0, $èv . . ., ^ ^ are all the classes and if we use the
same notation as above, it can be shown that © is equal to its commutator
group ©', if and only if the formula holds
Ä-1
(5)
f Ä A -
• # * - ! = Ugt.(K0
+ K1+...
+
K^)
where g^ is the number of elements in the class Kt. Thus the conjecture on
groups of odd order is equivalent with the fact that the equation (5) can never
hold for a group of odd order 7 ).
7
) I was informed by H. Wielandt that he also found this result.
216
Let me finish this talk with a number of remarks which are of some
interest in connection with our problems. Let ©x denote the set © — {1} obtained from © by removing the identity 1. Then the distance d(a, x) of two
elements of © t can be defined as follows.For a = x, we set d(a, x) = 0. If a and
r commute, but if o* ^ x, we set d(o,r)~ 1. If there exists a chain of elements
<r< of © 1;
a0 = a,ox, Ofc, . . ., ar = x,
of length r such that any two consecutive elements Gi_x and c^ commute, let
d(a, x) be the length of the shortest such chain. Finally, if no such chain exists,
set d(a, x) = oo. It is clear that ©x then becomes a metric space (in which
infinite distances are permitted).
Let © now be an arbitra^ group of even order g. The geometric part of the
proof of the characterization of PSL (3, q) shows that it will be of importance
in the general case to consider the set 3ft of involutions (or the set ©-J as a kind
of geometry in which the fundamental group is given by the transformations
Ta ' J -> 0~x]oi where a is a fixed element of © while / ranges over 3ft (or over
©x). A very first step in this direction is given by the consideration of the
distances. The following results can be proved without much difficulty.
Theorem. If a ^ 1 is a real element of a group © of even order and if the
distance d(a,3R) of a from the set of involutions is larger than 3, then the distance
d(a,W) is infinite. In this case, the normalizer 3l(cr) of a is an abelian group §
whose order is relatively prime to the index (© : §). The group § consists only of
real elements. It is the normalizer of each of its elements different from 1.
Theorem. If the group © of even order contains more than one class of involutions, then any two involutions of © have at most distance 3.
By combining these two statements, it is seen that if a group of even order
contains more than one class of involutions, then any two real elements or,
x ^ 1 of © either have infinite distance or distance d(a, x) 5j 9.
HARVARD UNIVERSITY.
217