Arch. Math. 96 (2011), 207–214
c 2011 The Author(s). This article is published
with open access at Springerlink.com
0003-889X/11/030207-8
published online February 19, 2011
DOI 10.1007/s00013-011-0226-5
Archiv der Mathematik
On finite groups whose Sylow subgroups have a bounded
number of generators
Colin D. Reid
Abstract. Let G be a finite non-nilpotent group such that every Sylow
subgroup of G is generated by at most δ elements, and such that p is the
largest prime dividing |G|. We show that G has a non-nilpotent image
G/N , such that N is characteristic and of index bounded by a function of
δ and p. This result will be used to prove that G has a nilpotent normal
subgroup of index bounded in terms of δ and p.
Mathematics Subject Classification (2010). 20D20.
Keywords. Finite group theory, Sylow theory.
1. Introduction. It will be assumed throughout that G is a finite group, and
Sp is a Sylow p-subgroup of G.
Definition 1.1. Write P for the set of all primes, and p for the set of all primes
except for p.
The Frattini subgroup Φ(G) of G is the intersection of all maximal subgroups of G. The Fitting subgroup F (G) of G is the largest nilpotent normal
subgroup of G. Note that Φ(G) ≤ F (G) for any finite group; see for instance
Theorem 5.2.15 of [8]. Also, a finite group is nilpotent if and only if it is the
direct product of its Sylow subgroups; it is this characterisation of nilpotency
that will play the key role in this paper.
Write d(G) for the minimum size of a generating set of G. Define
dp (G) :=
d(Sp ); equivalently, dp (G) = logp |Sp /Φ(Sp )|. Define σ(G) to be p∈P dp (G),
and δ(G) := maxp∈P dp (G). Define δ (G) as the largest integer δ such that
This paper is partly based on results obtained by the author while under the supervision
of Robert Wilson at Queen Mary, University of London (QMUL). My thanks go to Charles
Leedham-Green, László Pyber and Robert Wilson for their comments and advice, to the
referee for pointing out an error in the paper, and to EPSRC and QMUL for their financial
support.
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C. D. Reid
Arch. Math.
at least two primes p satisfy dp (G) ≥ δ . Define λ(G) to be the largest prime
dividing |G| (set λ(G) = 2 if G is trivial).
Unless otherwise indicated, we will save notation by writing δ for δ(G), δ for δ (G), σ for σ(G) and λ for λ(G).
Remark 1.2. This paper makes use of results derived from the classification
of finite simple groups (CFSG), namely Theorem 2.6 and its Corollary 2.7.
Theorem 4.1 includes the Odd Order Theorem but is based on results that
predate CFSG. Those assertions in Theorems A, B and C which depend on
these results are preceded with an asterisk.
Much of the theory of finite groups is built on the study of p-subgroups,
in particular Sylow subgroups, and the interactions between different primes.
In some situations, a bound on the number of generators of a Sylow subgroup
is significant. For instance, it was shown by Guralnick [3] and Lucchini [5]
that d(G) ≤ δ(G) + 1, for any finite group G. This paper concerns results in
a similar vein, with the hypothesis that bounds are given for σ(G) and δ(G),
together with the largest prime λ(G). Specifically, if G is not nilpotent, then
it has a non-nilpotent characteristic image of bounded order, and a nilpotent
characteristic subgroup of bounded index.
Theorem A. Let G be a non-nilpotent finite group. Let N be a subgroup that
is maximal subject to the conditions that N is characteristic and G/N is not
nilpotent. Then exactly one of the following holds.
(i) The quotient G/N is of the form S H, where S is an elementary abelian
p-group for some prime p and H is a nilpotent p -group that acts faithfully on S by conjugation. The order of G/N is at most pc1 δ /2, where
c1 = log 288/ log 9 < 8/3.
(ii) There is a characteristic subgroup of G/N that is the direct product of at
most δδ copies of a non-abelian finite simple group. *The order of G/N
2
is at most cλ2 δδ , where c2 is an absolute constant.
Theorem B. Let G be a finite group. *Then |G : F (G)| is at most the iterated
...µ
2
exponential μμ , where μ = cλ3 δδ for c3 an absolute constant.
σ
The following generalisation of Theorem B to profinite groups strengthens
a result of Mel’nikov [6].
Corollary 1.3. Let Γ be a profinite group, such that for every prime p, every
maximal pro-p subgroup of Γ is topologically generated by at most d elements,
and such that no prime dividing the order of a finite continuous image of Γ
...µ
2 2
exceeds n. Set μ = cn3 d and set ν = μμ . Then Γ has a pro-(finite nilpotent)
nd
closed normal subgroup N such that |Γ : N | ≤ ν.
Finally, it can be concluded from Theorem B that the number of non-abelian composition factors of G is bounded by a function of δ(G) and λ(G).
In fact, a bound on the number of non-abelian composition factors can be
obtained with weaker conditions on G, using a different method of proof.
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Sylow subgroups with bounded number of generators
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Theorem C. Let G be a finite group and let p and q be distinct primes. Then
the number of composition factors of G of order divisible by pq is bounded by a
function of (p, q, dp (G), dq (G)). *The total number of non-abelian composition
factors of G is bounded by a function of (d2 (G), d3 (G), d5 (G)).
Remark 1.4. There is no equivalent to Theorem C if we only consider dp (G)
for a single prime: for instance, if G = Ar Cpn where Cpn is the cyclic group
of order pn with regular action, r = max{p, 5} and Ar is the alternating group
of degree r, then dp (G) ≤ 3, but G has pn non-abelian composition factors of
order divisible by p.
2. Preliminaries
Definition 2.1. Given a set of primes π, write Oπ (G) be the smallest normal
subgroup of G such that G/Oπ (G) is a π-group, and write Oπ (G) for the largest normal π-subgroup of G. Note that G is nilpotent if and only if Op (G) is
a p -group for every prime p.
The following theorem plays a critical role in the proofs in this paper.
It follows from work of Thompson and of Huppert, and was later proved in
somewhat greater generality than given here by Tate, who also gave a shorter
proof.
Theorem 2.2 (Thompson (unpublished); Huppert [4]; Tate [11]). Let G be a
finite group, and let Sp be a Sylow p-subgroup of G for some prime p. Let
N G such that N ∩ Sp ≤ Φ(Sp ) for some prime p. Then Op (N ) is a p -group.
Corollary 2.3. (i) Let M G and N G such that N ≤ M and M ∩ Sp ≤
Φ(Sp )N for some prime p. Then Op (M/N ) is a p -group. In particular,
if M ∩ Sp ≤ Φ(Sp )N for every p then M/N is nilpotent.
(ii) Let N G such that N ∩ Sp ≤ Φ(Sp ) for every p. Then F (G/N ) =
F (G)/N .
Proof. (i) This follows immediately from the theorem, noting that Sp N/N is
a Sylow p-subgroup of G/N , and that Φ(Sp N/N ) = Φ(Sp )N/N .
(ii) By (i), N is nilpotent, so N ≤ F (G). Let T be the lift of F (G/N ) to G. It
remains to show that T is nilpotent, or equivalently that Op (T ) is a p -group
for every prime p. Certainly Op (T )N/N is a p -group, as T /N is nilpotent.
But then Op (T ) ∩ Sp ≤ N ∩ Sp ≤ Φ(Sp ), so Op (T ) = Op (Op (T )) is a p -group
by the theorem.
We will need bounds on the orders of nilpotent groups in terms of certain
faithful actions.
Theorem 2.4 (Wolf [14]). Let G be a nilpotent subgroup of GL(n, pe ) of order
coprime to p. Then logp (2|G|) ≤ en log 32/ log 9.
Theorem 2.5 (Vdovin [15]). The order of a nilpotent subgroup of Sym(n) is at
most 2n .
For case (ii) of Theorem A, a bound is required on the order of the simple subnormal subgroups of G/N . For the order of simple factors, a bound
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C. D. Reid
Arch. Math.
emerges from λ(G). Indeed, we can use a bound on the order of the automorphism groups induced on such factors, as |Out(G)| < |G| for any finite simple
group G (see for instance Lemma 2.2 of [7]).
Theorem 2.6 (Babai, Goodman, and Pyber [1]: Theorem 5.4 and subsequent
remark). Suppose G is a finite simple group whose order has no prime divisor
2
greater than some positive integer k. Then |G| < ck for an absolute constant c.
2
Corollary 2.7. Let G be a finite simple group. Then |Aut(G)| < cλ(G) for an
absolute constant c.
For the number of non-abelian composition factors, on the other hand, the
critical constraint for our purposes is the number of generators of the Sylow
subgroups.
Lemma 2.8. Let p be a prime, let Ω be the set of subnormal non-abelian simple
subgroups of G whose order is divisible by p, and let n be the number of orbits
of Sp acting on Ω by conjugation. Then n ≤ dp (G).
Proof. We may assume that G = KSp , where K = Ω. Since |Sp : Φ(Sp )| =
pdp (G) , there is a subset Ξ of Ω such that |Ξ| ≤ dp (G) and such that K ∩ Sp ≤
N Φ(Sp ), where N is the normal closure of Ξ in G. By Corollary 2.3 it follows
that Op (K/N ) is a p -group. As K is perfect, in fact Op (K/N ) = K/N , and
since p divides the order of every composition factor of K, we conclude that
K = N . It follows that given Q ∈ Ω, there is some x ∈ G such that Qx ∈ Ξ;
since K normalises Q, there is a suitable x in Sp . Hence n ≤ |Ξ| ≤ dp (G) as
required.
For Theorem B we will need a modified form of the Schreier index formula.
Lemma 2.9. Let G be a finite group, and let H be a subgroup. Then
δ(H) ≤ |G : H|δ(G);
δ(H)δ (H) ≤ |G : H|δ(G)δ (G).
Proof. Let T be a p-Sylow subgroup of H, contained in a p-Sylow subgroup
S of G. Then by the Schreier index formula, d(T ) − 1 ≤ np (d(S) − 1) where
np = |S : T | is the largest power of p dividing |G : H|. We see from this that
δ(H)/δ(G) is at most np for some p, while δ(H)δ (H)/(δ(G)δ (G)) is at most
np nq for distinct primes p and q; in turn, np ≤ np nq ≤ |G : H|.
3. The main theorems
Proof of Theorem A. We may assume that G = G/N , so that for every nontrivial characteristic subgroup K of G, the image G/K is nilpotent. Note that
G has a unique minimal characteristic subgroup M , for if M1 and M2 were
two such, then the non-nilpotent group G would be isomorphic to a subgroup
of the nilpotent group G/M1 × G/M2 , which is impossible.
Suppose that M is soluble. Then M is an elementary abelian p-group for
some p. It follows that G has a unique Sylow p-subgroup S say, as G/M is nilpotent. By the Schur-Zassenhaus theorem, G is of the form S H, where H is a
Vol. 96 (2011)
Sylow subgroups with bounded number of generators
211
p -group; note that H ∼
= G/S, so H is nilpotent. Furthermore, F (G/Φ(S)) =
F (G)/Φ(S) < G/Φ(S) by Corollary 2.3, so G/Φ(S) is not nilpotent; thus
Φ(S) = 1. So S is elementary abelian, and logp (|S|) = d(S) ≤ δ.
Note that CS (H) is characteristic in G (as CS (H) = Op (G)) and does
not contain M , so CS (H) = 1. It follows that H is isomorphic to a nilpotent
p -subgroup of GL(δ, p); hence |H| ≤ p(c1 −1)δ /2 by Theorem 2.4, so |G| =
|H||S| ≤ pc1 δ /2. We have now proved all of the assertions for case (i).
Now suppose that M is insoluble. Then M is a direct product of a set
Ω = {Q1 , . . . , Qk } of isomorphic copies of a non-abelian finite simple group Q.
Let G act on Ω by conjugation and let p and q be two distinct primes which
divide the order of Q; we may assume dq (G) ≤ δ . Then the number of orbits
of Sp on Ω is at most δ by Lemma 2.8; moreover, the orbits all have the same
size, say pm , since M acts trivially on Ω and Sp M is characteristic in G. Thus
|Ω| = xpm for some non-negative integers x and m such that x ≤ δ; similarly
|Ω| = x q m for some x and m such that x ≤ δ . This produces an upper
bound for |Ω| as follows:
|Ω| = gcd(|Ω|, |Ω|) ≤ gcd(xpm , x ) gcd(xpm , q m ) ≤ x x ≤ δδ .
k
Let R = i=1 NG (Qi ). Then G/R is isomorphic to a nilpotent subgroup of
Sym(Ω), so |G/R| ≤ 2δδ by Theorem 2.5.
2
By Corollary 2.7 we have |Aut(Q)| < cλ where c is an absolute constant.
Since CG (M ) is characteristic and M ∩ CG (M ) = 1, we also have CG (M ) = 1,
2
2
so that |R| ≤ |Aut(Q)|δδ . Hence |G| ≤ (2cλ )δδ ≤ (21/25 c)δδ λ . We have now
proved all of the assertions for case (ii).
An obstacle to obtaining a bound under weaker conditions is illustrated by
the following example.
Example 3.1. (My thanks go to Robert Wilson for suggesting this example.)
e
Let G = SL(2, q) t, where q = pp and t acts as the Frobenius automorphism f : x → xp of the defining field Fq . The order of t is pe , and a Sylow
p-subgroup of G is generated by t together with the element s of SL(2, q),
where
1 ν
s=
0 1
such that ν is a primitive element of Fq . If p is odd then the 2-Sylow subgroups
of SL(2, q) and hence of G are generated by 2 elements (see [2]) and if r is an
odd prime not equal to p then the r-Sylow subgroups of SL(2, q) and hence of
G are cyclic (see [13]).
Thus for any values of p and e we have δ(G) = 2. On the other hand,
given a non-nilpotent image G/N of G, then N is a proper normal subgroup
of SL(2, q), and hence |N | ≤ 2. In other words, every non-nilpotent image of
e
e
G has order at least pe q(q 2 − 1)/2 = pp +e (p2p − 1)/2.
The non-nilpotent images described in Theorem A will now be used to
prove Theorem B and its application to profinite groups.
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C. D. Reid
Proof of Theorem B. We may assume that |G| is not a prime power, so that
δ ≥ δ ≥ 1. Choose a finite sequence Gi of subgroups as follows: G0 = G, and
if Gi is not nilpotent, then Gi+1 is a subgroup of Gi that is maximal subject to
the conditions that Gi+1 is characteristic in Gi and Gi /Gi+1 is non-nilpotent.
The sequence terminates with a nilpotent characteristic subgroup Gt for some
t ≥ 0. As F (G)
≥ Gt , it suffices to obtain a suitable bound on |G : Gt |.
Let ri = p∈P logp |(Gi ∩ Sp )Φ(Sp )/Φ(Sp )|. Then r0 = σ, and ri ≥ ri+1 for
all i. Suppose ri = ri+1 for some i < t. Then (Gi ∩Sp )Φ(Sp ) = (Gi+1 ∩Sp )Φ(Sp )
for all p ∈ P, so Gi /Gi+1 is nilpotent by Corollary 2.3, contradicting the choice
of Gi+1 . Hence r0 , . . . , rt is a strictly decreasing sequence of non-negative integers, bounded above by σ; thus t ≤ σ.
Let c1 and c2 be as in Theorem A, and set c3 = max{2, 21/25 c2 }. Let g0 = 1
2
λ
. Suppose |G : Gi | ≤ gi for some
and thereafter gi+1 = μgi , where μ = cδδ
3
integer i. Then δ(Gi ) ≤ δgi and δ(Gi )δ (Gi ) ≤ δδ gi by Lemma 2.9. It follows
from Theorem A that if Gi /Gi+1 is soluble, the index |G : Gi+1 | is at most
gi λc1 δgi /2, whereas if Gi /Gi+1 is insoluble, the index |G : Gi+1 | is at most
2
λ gi
gi cδδ
. In the first case we use the facts that c1 < 8/3 and λ ≥ 3, in the
2
second that λ ≥ 5, and in both cases we use the fact that n < 2n for any
integer n:
2
gi λc1 δgi /2 < 2(1+8λδ/3)gi ≤ 2λ
2
λ gi
gi cδδ
2
<
2
λ gi
2gi cδδ
2
≤
δgi
;
2
2
λ gi
2λ gi /25 cδδ
2
2
≤ (21/25 c2 )δδ λ
gi
.
We conclude in both cases that
2
λ
|G : Gi+1 | < cδδ
3
gi
= gi+1 .
Thus |G : Gt | is at most gt by induction, which is at most gσ as required.
Remark 3.2. In [1], the authors conjecture an upper bound of 2 for the absolute
constant c appearing in Theorem 2.6. If this conjecture is correct, we can specify log2 (c2 ) ≤ 2.04 in Theorem A and log2 (c3 ) ≤ 2.08 < log2 (5) in Theorem B.
Proof of Corollary 1.3. Let K be an open normal subgroup of G. The hypotheses ensure that δ(G/K) ≤ d, λ(G/K) ≤ n and σ(G) ≤ dn; thus by Theorem B,
|G/K : F (G/K)| ≤ ν. Let NK be the lift of F (G/K) to G, and let N be the
intersection of the NK as K ranges over all open normal subgroups. Then
|G : N | ≤ ν. Moreover, N is closed in G, and is the inverse limit of finite
nilpotent groups.
4. A bound on the number of non-abelian composition factors. To obtain a
bound on the number of non-abelian composition factors, with weaker hypotheses than required for Theorem B, we will make use of some more known
results.
Theorem 4.1. Let G be a non-abelian finite simple group. Then |G| is divisible
by at least one of 6 and 10.
Proof. This follows from Corollary 1 of [12], which gives all non-abelian simple
groups whose proper subgroups are soluble: all such groups have order divisible
by 6, except Sz(2p ) which has order divisible by 10.
Vol. 96 (2011)
Sylow subgroups with bounded number of generators
213
Definition 4.2. Given x ∈ N, let sp (x) be the sum of the digits of the base-p
expansion of x.
Theorem 4.3 (Senge and Straus [9]; see also Stewart [10]). Let p and q be
distinct primes and let n ∈ N. Then the set {x ∈ N | sp (x) ≤ n, sq (x) ≤ n} is
finite.
Proof of Theorem C. By dividing out by a suitable normal subgroup as necessary, we may assume that every non-trivial normal subgroup of G has a
composition factor of order divisible by pq. Let Ω be the set of subnormal nonabelian simple subgroups of G, let x = |Ω|, and let K = Ω. By Lemma 2.8, Sp
has at most dp (G) orbits on Ω; since each orbit has p-power order, this forces
sp (x) ≤ dp (G). Similarly, sq (x) ≤ dq (G). It follows from Theorem 4.3 that x
is bounded by a function of (p, q, dp (G), dq (G)). Moreover, K = Op (K) is not
a p -group, so K ∩ Sp ≤ Φ(Sp ) by Theorem 2.2; thus dp (G/K) < dp (G). It follows by induction on dp (x) that the total number of non-abelian composition
factors of G divisible by pq is bounded by a function of (p, q, dp (x), dq (x)).
The final assertion now follows immediately from Theorem 4.1.
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are credited.
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Colin D. Reid
Mathematisches Institut,
Georg-August Universität Göttingen,
Bunsenstraße 3-5, 37073 Göttingen,
Germany
e-mail: [email protected]
Received: 22 September 2010
Revised: 18 November 2010