On the indices of maximal subgroups and the normal primary coverings of finite groups

Francesco Fumagalli

doi:10.1515/jgth-2018-0212

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On the indices of maximal subgroups and the normal primary coverings of finite groups

Published/Copyright: May 7, 2019

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From the journal Journal of Group Theory Volume 22 Issue 6

Abstract

We define and study two arithmetic functions γ0 and η, having domain the set of all finite groups whose orders are not prime powers. Namely, if G is such a group, we call γ0⁢(G) the normal primary covering number of G; this is defined as the smallest positive integer k such that the set of primary elements of G is covered by k conjugacy classes of proper (pairwise non-conjugate) subgroups of G. Also we set η⁢(G), the indices covering number of G, to be the smallest positive integer h such that G has h proper subgroups having coprime indices. This second function is an upper bound for γ0, and it is much friendlier. The study of these functions for arbitrary finite groups reduces immediately to the non-abelian simple ones. We therefore apply CFSG to obtain bounds and interesting properties for γ0 and η. Open questions on these functions are reformulated in pure number-theoretical terms and lead to problems concerning the distributions and the representations of prime numbers.

Introduction

A finite non-trivial group G can never be the set-theoretic union of all the conjugates of a fixed proper subgroup H. For the question of whether there are elements outside ⋃g∈GHg possessing some interesting group theoretical properties, a significant answer is given in [9, Theorem 1]. The authors prove that, for any H<G, there always exists an element of prime power order which does not lie in ⋃g∈GHg. An equivalent statement is that every non-trivial finite transitive permutation group has a derangement (i.e., an element acting fixed-point-free) of prime power order. This result – which relies on the Classification of Finite Simple Groups – permits the authors to prove that there is no global field extension 𝕃⊇𝕂 such that the reduced Brauer group B⁢(𝕃/𝕂) is finite, a theorem of significant importance in algebraic number theory.

In this paper, given a finite group G, whose order is not a prime power, we let G0 be the set of primary elements of G (we recall that a primary element of a group is an element having prime power order) and, along the lines of [3, 7, 2, 4, 5], define a normal primary covering for G to be any collection of complete conjugacy classes of subgroups of G whose union contains G0. The normal primary covering number of G is by definition the cardinality of a minimal normal primary covering, namely, the smallest natural number γ0⁢(G) such that

G0⊆⋃i=1γ0⁢(G)⋃g∈GHig

for some proper – pairwise non-conjugate – subgroups Hi of G.

Theorem 1 in [9] can therefore be reformulated by saying that γ0⁢(G)≥2.

We are interested in bounding the function γ0. For this, we define a second function on the set of finite groups G, whose orders are not prime powers, by setting η⁢(G) to be the smallest number of proper subgroups of G having coprime indices. We call η⁢(G) the indices covering number of G. Note that η⁢(G) is always an upper bound for γ0⁢(G). It happens that this second function is much friendlier than γ0. Most of the paper is addressed to the study of η, which immediately reduces to the case of finite non-abelian simple groups (Proposition 1). We therefore make use of the Classification theorem.

If, following the number-theoretic literature (for instance, [20]), we denote with ω⁢(n) the number of distinct prime divisors of n; our results on alternating groups can be summarized in the following theorem.

Theorem A.

If An is the alternating group of degree n≥5, then

η⁢(An)≤ω⁢(n)+1 (Lemma 2),
η⁢(An)=2 if and only if n is a prime power (Theorem 1),
lim infω⁢(n)→∞⁡η⁢(An)=3 (Theorem 2).

For groups of Lie type, we manage to prove the following (Theorems 3 and 4):

Theorem B.

If G is a simple group of Lie type of rank n, then η⁢(G) is bounded above by a linear function in ω⁢(n). In particular, if G is an exceptional group, η⁢(G) is uniformly bounded.

Tables 1, 2 and 3 provide upper bounds (which may be close to best possible) for the classical, the exceptional and the sporadic simple groups, respectively.

The last part of the paper is devoted to normal primary coverings and the related function γ0. In particular, we show that γ0 and η are in general different functions (Proposition 6).

The paper leaves open the following questions.

Question A.

Do there exist positive constants C and D such that, for every finite group G (whose order is not a prime power), γ0⁢(G)≤C and η⁢(G)≤D?

Of course, if η is uniformly bounded, then so is γ0. Our analysis allows us to reduce the above question (relative to the case of alternating groups) to problems concerning distributions and representations of prime numbers. In particular, Question A for the function η and when G is alternating, can be restated in the following purely number-theoretical form.

Question B.

Given a positive integer n, let η⁢(n) be the smallest number such that

gcd⁡{(nm1),…,(nmη⁢(n))}=1if⁢n⁢is odd,

gcd⁡{(nm1),…,(nmη⁢(n)),12⁢(nn/2)}=1if⁢n⁢is even,

for some 1≤mi<n/2 for every i. Is it true that lim sup⁡η⁢(n)<+∞?

1 Some basic results for the function η⁢(G)

As stated in the introduction, for every finite group G whose order is not a prime power, we define the indices covering number of G, η⁢(G), to be the smallest number of proper subgroups of G having coprime indices, i.e., η⁢(G)=k if and only if there are k (and not fewer) proper subgroups of G, say, H0,H1,…,Hk-1, with the property

(CI)gcd(|G:H0|,|G:H1|,…,|G:Hk-1|)=1.

The following lemma collects some elementary properties. We recall that, following the number-theoretic literature [20], we set ω⁢(n) for the number of distinct prime divisors of a positive integer n. The proof of the lemma is left to the reader.

Lemma 1.

Let G be a finite group whose order is not a prime power. Then the following holds.

η⁢(G)≤ω⁢(|G|).
η⁢(G) can always be realized by taking a collection of maximal subgroups Hi of G (for i=0,1,…,η⁢(G)-1).
If η⁢(G)=2, then G=H0⁢H1 (Poincaré lemma).
If N is a proper normal subgroup of G, then η⁢(G)≤η⁢(G/N), and if G/N is a p-group, for some prime p, then η⁢(G)=2.

The following proposition is easy but crucial.

Proposition 1.

Let G be a finite group whose order is not a prime power. Then η⁢(G)=2 if G′≠G. In particular, if G is a soluble group or an almost simple group which is not simple, then η⁢(G)=2.

Proof.

By Lemma 1 (4) with N=G′, it is enough to show that the result holds for G a finite abelian group of non-prime power order. In this case, choose a prime p dividing |G|, and take as H0 a (maximal) subgroup containing the Sylow p-subgroup P of G and as H1 a maximal one containing the complement of P in G. ∎

In virtue of Proposition 1, the study of the function η takes its interest only for perfect groups and, in particular, it immediately reduces to the case of finite non-abelian simple groups. However, the following remark shows that there are perfect groups G having η⁢(G)<η⁢(G/N), for every proper non-trivial normal subgroup N.

Remark 1.

Let G=A6×PSL2⁢(13). By looking at the list of maximal subgroups (for instance, in [6]), it is immediate to see that η⁢(A6)=η⁢(PSL2⁢(13))=3. However, G has maximal subgroups isomorphic to A6×A4 and A5×PSL2⁢(13), and therefore η⁢(G)=2.

From now on, we assume that G is a finite non-abelian simple group. We separately treat the various cases according to CFSG.

2 The case of alternating groups

Let An be the alternating group of degree n≥5 that is acting on the natural set {1,2,…,n}. We start with this simple lemma.

Lemma 2.

η⁢(An)≤ω⁢(n)+1.

Proof.

Take H0=StabAn⁡(1) to be the stabilizer of point 1. Then H0≃An-1 has index n in An. For every prime pi dividing n, let Hi be a maximal subgroup containing a Sylow pi-subgroup of An. Then the collection of subgroups

ℋ={H0,H1,…,Hω⁢(n)}

has property (CI), and therefore η⁢(An)≤ω⁢(n)+1. ∎

Lemma 2 in particular implies η⁢(An)=2 whenever n is a prime power. We will prove that the converse of this statement holds (see Theorem 1).

From now on, assume that the degree n is not a prime power.

Moreover, we make the following important assumption. We always include a one-point stabilizer (as subgroup H0) in a list of maximal subgroups of An whose cardinality is at least η⁢(An)+1, meaning that if η⁢(An)=k and there is a list ℋ={H1,H2,…,Hk} of maximal subgroups of coprime indices and not containing the stabilizer of a single point, then we consider the extended list ℋ^=ℋ∪{H0}. The family ℋ^ still has property (CI). The advantage of this choice is evident: the computation of η⁢(An) is reduced to finding which maximal subgroups of An contain Sylow p-subgroups for the various primes p that divide n.

The maximal subgroups of An split in three different classes, according to their action on {1,2,…,n}: intransitive subgroups, imprimitive and primitive ones.

The primitive maximal subgroups are unnecessary for the computation of the indices covering number; this is a consequence of the following famous result of C. Jordan.

Lemma 3.

No proper primitive subgroup of An contains p-cycles for primes p, where p≤n-3. In particular, no primitive maximal subgroup of An contains a Sylow p-subgroup for p a proper divisor of n.

Proof.

We refer to [24, Theorem 13.9]. ∎

Before considering in detail the other cases of maximal subgroups of An, we introduce some more notation.

Given two natural numbers n and b≥2, if

n=n0+n1⁢b+n2⁢b2+⋯+nk⁢bk

is the expansion of n in base b, we write

[n]b=(n0,n1,…,nk).

Also, if m and d are other positive numbers, we write

[n]b≤[m]bif⁢ni≤mi,

[n]b=d⋅[m]bif⁢ni=d⋅mi

for all i=0,1,…,max⁡{⌊logp⁡(n)⌋,⌊logp⁡(m)⌋}. We recall also that if p is a prime, the p-adic value of n!, namely, the exponent of the p-part of n!, is usually denoted by vp⁢(n!) and is given by

(2.1)vp⁢(n!)=∑i≥1⌊n/pi⌋

and, equivalently, if [n]p=(n0,n1,…,nk), then

(2.2)vp⁢(n!)=∑i=1kni⋅pi-1p-1

(see, for instance, [20, Theorem 1.12] or [8, Example 2.6.8]).

Consider now the case of intransitive maximal subgroups of An. Any such subgroup is the setwise stabilizer of a set of cardinality m, for some 1≤m<n/2, being therefore isomorphic to (Sm×Sn-m)∩An and having index (nm). For convenience, we choose a prototype of these subgroups by letting Xm be the stabilizer of the set {1,2,…,m} in An.

The next lemma says exactly when a maximal intransitive subgroup contains a Sylow p-subgroup of An, for some prime number p which divides n. This is basically a result of Ernst Kummer [15, pp. 115–116]. See also [21, Lemma 3.1] and [11] for an overview.

Lemma 4.

Let 1≤m<n/2 and p a prime number, p≤n. The following conditions are equivalent.

p does not divide (nm);
Xm contains a Sylow p-subgroup of An;
any p-element of An lies in a conjugate of Xm;
[m]p≤[n]p.

Proof.

Since |An:Xm|=|Sn:Sm×Sn-m|=(nm), we prove the analogous statement for the groups Sm×Sn-m≤Sn instead of Xm≤An.

The implications (1) ⇒ (2) ⇒ (3) are trivial consequences of Sylow theorems. We prove (3) ⇒ (4). Write the p-adic expansions of n and m respectively as

[n]p=(n0,n1,…,nl) and [m]p=(m0,m1,…,ml),

where l=⌊logp⁡n⌋ and mi=0 for i=⌊logp⁡m⌋+1,…,l. The group Sn contains p-elements whose cycle type consists of exactly ni cycles of length pi, for i=1,2,…,l. By (3), let g be such an element of Xm. The set {1,2,…,m} being the union of 〈g〉-orbits, we have mi≤ni for every i=0,1,…,⌊logp⁡(n)⌋, i.e., [m]p≤[n]p.

Finally, assuming (4), from formula (2.2), it is straightforward to see that

vp⁢(n!)=vp⁢(m!)+vp⁢((n-m)!),

which is equivalent to (1). ∎

Corollary 1.

If any of the conditions (1)–(4) of Lemma 4 is verified, then the p-part of n divides both m and n-m.

We consider now the case of imprimitive maximal subgroups. Any such subgroup of An is the stabilizer of a partition of {1,2,…,n} into equal-sized subsets, and therefore it is isomorphic to the wreath product (Sd≀Sn/d)∩An, for some proper non-trivial divisor d of n, its index in An being

n!(d!)n/d⋅(n/d)!=:In,d.

As before, for convenience, we set Wd to be the stabilizer in An of the partition

{{1,2,…,d},{d+1,d+2,…,2⁢d},…,{n-d+1,n-d+2,…,n}}.

We prove the analogous result of Lemma 4, which already appeared in [23, Lemma 2] (see also [21, Lemma 3.2]).

Lemma 5.

Let d be a proper non-trivial divisor of n. Then p∤In,d, i.e., the imprimitive maximal subgroup Wd contains a Sylow p-subgroup of An if and only if one of the following is satisfied:

either d is a p-power, or
n/d<p and [n]p=n/d⋅[d]p.

Proof.

For convenience, we set l=n/d. Since In,d=|Sn:Sd≀Sl|, we prove the lemma by considering the subgroup Hd:=Sd≀Sl of Sn, instead of Wd in An.

We first assume that Hd contains a p-Sylow subgroup of Sn, equivalently that vp⁢(n!)=l⋅vp⁢(d!)+vp⁢(l!).

As d>1, p divides n!l!, and therefore vp⁢(n!)>vp⁢(l!), showing that vp⁢(d!)>0, i.e., d≥p. We write

[n]p=(n0,n1,…,nu),

[d]p=(d0,d1,…,dh),

[l]p=(l0,l1,…,lk).

Note that u=⌊logp⁡n⌋ is either h+k or h+k+1. If u=h+k+1, the group Sn contains cycles of length ph+k+1. Note that none of these p-elements belong to Hd since, as ph+k+1>d, the support of such an element must meet at least pk+1 blocks, which do not exist. Thus u=h+k. Assume now that k≥1, equivalently that l≥p. Consider a cycle of length ph+k that lies in Hd. Such an element would cyclically permute pk blocks, and its support must be a union of complete blocks; this shows that d divides ph+k, i.e., condition (i) holds. Now let k=0, and show (ii), i.e., ni=l⋅di for any i=0,1,…,u. Arguing by contradiction, we set r to be the largest integer for which nr>l⋅dr (of course, nu≥l⋅du). The full symmetric group Sn contains p-elements which are products, for i running from r to u, of ni disjoint cycles of length pi each. However, for similar reasons as before, such an element cannot stay in Hd, which contradicts the fact that Hd contains a Sylow p-subgroup. Therefore, we proved

(2.3)ni≤l⋅di for every⁢i=0,1,…,u.

But then nu=l⋅du and, from this and equation (2.3), it is then straightforward to prove ni=l⋅di for every i=0,1,…,u.

Conversely, first assume condition (i), and let d=pr, so that the p-adic expansions of n and l=n/d are, respectively,

[n]p=(0,…,0,nr,…,nu)

[l]p=(nr,…,nu),

where the first line starts with r zeros, n being divisible by pr.

We have

l⋅vp⁢(d!)+vp⁢(l!)=nd⋅pr-1p-1+∑i=0u-rnr+i⁢pi-1p-1=∑i=0u-rnr+i⁢pi⋅pr-1p-1+∑i=0u-rnr+i⁢pi-1p-1=∑i=0u-rnr+i⁢(pi⁢pr-1p-1+pi-1p-1)=∑i=0u-rnr+i⁢(pr+i-1p-1)=vp⁢(n!).

Now assume (ii), and write [d]p=(d0,…,dh) and [n]p=(l⁢d0,…,l⁢dh). Thus

l⋅vp⁢(d!)+vp⁢(l!)=l⋅∑i≥1di⁢pi-1p-1=∑i≥1l⋅di⁢pi-1p-1=vp⁢(n!),

which completes the proof. ∎

Lemma 5 has these easy but important consequences.

Corollary 2.

Let d be a non-trivial proper divisor of n.

If d is a p-power, then Wd contains Sylow p-subgroups of An, but not Sylow q-subgroups for any other prime q≠p dividing n.
If Wd contains a Sylow p1-subgroup, with p1 the smallest prime divisor of n, then d is a p1-power and Wd does not contain any other Sylow p-subgroup for p≠p1 dividing n.

Proof.

(1) Let d=pr, and let q be a different prime dividing n. Then q divides n/d, forcing n/d≥q. By the previous lemma, Wd cannot contain a Sylow q-subgroup of An.

(2) If d is not a power of p1 and Wd contains a Sylow p1-subgroup of An, by Lemma 5, we have n/d<p1, which is a contradiction since p1 is the smallest prime divisor of n. ∎

Proposition 2.

Let n be different from a prime power. If n is odd, the maximal intransitive subgroups of An are enough to produce a set of (maximal) subgroups of An realizing η⁢(An). If n is even, there always exists a set of (maximal) subgroups realizing η⁢(An) consisting either entirely of intransitive subgroups or of intransitive subgroups and the subgroup Wn/2.

Proof.

Let ℋ be a collection of maximal subgroups realizing η⁢(An). Assume that ℋ contains some primitive maximal subgroup R. By Lemma 3, the only prime divisors of |An| that do not divide the index of R are greater than or equal to n-2. Since n is not a prime power, we may replace this subgroup R with the intransitive subgroup X1 in ℋ.

Assume now that ℋ contains some imprimitive maximal subgroup Wd, for some proper non-trivial divisor d of n. By Lemma 5, if d is not a prime power, the only primes r not dividing In,d are those for which n/d<r and [n]r=n/d⁢[d]r. Note that this implies [d]r≤[n]r, and so, by Lemma 4, these primes r do not divide (nd). As long as n≠2⁢d, we may therefore substitute Wd with Xd in ℋ. Suppose now that d is a prime power, say, d=pa. If pa+1∣n, then note that, in virtue of Lemma 5, the primes r not dividing In,d also do not divide In,d⁢p; therefore, the subgroup Wd⁢p may take the place of Wd in ℋ. Without loss of generality, we assume therefore that pa+1 does not divide n. Of course, Wd contains Sylow p-subgroups and possibly Sylow r-subgroups for those primes r such that n/d<r and [n]r=n/d⁢[d]r. As before, by Lemma 4, neither p nor any of these r’s divide (nd), and again if n≠2⁢d, the subgroup Xd can substitute Wd in the list ℋ. ∎

Example 1 below shows that, in some situations, the presence of Wn/2 is necessary for ℋ to realize η⁢(An).

We can now prove the already anticipated characterization of alternating groups having indices covering number equal to two.

Theorem 1.

Let n≥5. Then η⁢(An)=2 if and only if n is a prime power.

Proof.

One implication is obvious from the remark after Lemma 2.

Let us assume that η⁢(An)=2 and prove that n is a prime power. A direct inspection of the Atlas [6] shows that η⁢(A6)=η⁢(A10)=3; therefore, we assume n≥12. Let A and B be two proper subgroups of coprime indices; in particular, we have An=A⁢B. According to [16, Theorem D] up to changing A with B, we have A≃Xm and that B is m-homogeneous (i.e., transitive on the set of m-subsets of {1,2,…,n}), for some 1≤m≤5. By [8, Theorem 9.4B], if m≥2, then either B is 2-transitive or m=2 and A⁢S⁢L1⁢(q)≤B≤A⁢Γ⁢L1⁢(q), with n=q≡3(mod4), acting in the usual permutation representation on the field of q elements. Since, in the latter case, we clearly have that n is a prime power, we assume that B is 2-transitive and therefore primitive. Now B contains a Sylow p-subgroup of An for any p dividing (nm), the index of A. Let p1 be the smallest of these primes. If n-p1≥3, we reach a contradiction by Lemma 3. Therefore, n-p1≤2, and the only possible situations arise when n=7, p1=5 and m=3 or 4, which is not the case as n≥12. Therefore, m=1, A≃X1, and B is a transitive subgroup whose index is coprime with n. By Lemma 3, B is not primitive. Then B is a transitive imprimitive subgroup isomorphic to Wd, for some proper non-trivial divisor d of n. By Corollary 2, n is a prime power. ∎

In Lemma 2, we proved η⁢(An)≤ω⁢(n)+1. The following theorem shows that the difference between η⁢(An) and ω⁢(n) can be arbitrarily big.

Theorem 2.

lim infω⁢(n)→∞⁡η⁢(An)=3.

We need a preliminary lemma in number theory.

Lemma 6.

Let m=p1⁢p2⁢…⁢pk be the product of k distinct prime numbers pi. For every i=1,…,k, let ai be the pi-adic value of m!, and choose any prime number q which is congruent to 1 modulo p1a1⁢…⁢pkak. Then

(m⁢qm)≡1(modpi) for every⁢i=1,…,k.

Proof.

Let p be any prime from the set {pi}i=1k. By applying Lucas’ theorem (see, for instance, [11]) we have

(m⁢qm)≡∏j=0h((m⁢q)jmj)(modp),

where

[m]p=(m0,m1,…,mh),

[m⁢q]p=((m⁢q)0,(m⁢q)1,…,(m⁢q)h).

By our choice of q, we have m⁢q≡m(modp1a1⁢…⁢pkak) and (m⁢q)j=mj for every j=0,1,…,⌊logp⁡m⌋, while (m⁢q)j≥mj=0 for j=⌊logp⁡m⌋+1,…,h. This implies ((m⁢q)jmj)=1 for all j, and so (m⁢qm)≡1(modp). ∎

Proof of Theorem 2.

Let m and q be chosen as in the previous lemma, and let n=m⁢q. Of course, ω⁢(n)=k+1→∞ when k→∞.

By Lemma 6, the maximal subgroups Xm of Am⁢q contain Sylow p-subgroups for every prime p different from q that divides n. Therefore, the family

ℋ={X1,Xm,Q},

where Q is any Sylow q-subgroup of An, is made of subgroups of coprime indices, proving that η⁢(An)≤3. Finally, Theorem 1 completes the proof. ∎

Question A stated in the introduction, for the function η and the case of alternating groups becomes:

Question A${}^{\prime}$.

Is lim supω⁢(n)→∞⁡η⁢(An) always finite?

Our analysis permits the following reformulation in terms of pure number theory.

Question B.

Given a positive integer n, let η⁢(n) be the smallest number such that

gcd⁡{(nm1),…,(nmη⁢(n))}=1if⁢n⁢is odd,

gcd⁡{(nm1),…,(nmη⁢(n)),12⁢(nn/2)}=1if⁢n⁢is even,

where 1≤mi<n/2 for every i. Is it true that lim sup⁡η⁢(n)<+∞?

This seems to be quite a hard problem since it deals with the distributions and the representations (in different bases) of prime numbers. We made some computations with a program, considering values of n that are products of the first k distinct primes, up to k=10 (and so n≤6 469 693 230). Our data suggest that it might be η⁢(An)≥⌊k/2⌋-1, but more evidence should be gathered.

The following example shows a particular situation when n is even.

Example 1.

Consider n=826 610=2⋅5⋅131⋅631 and d=n/2. Then we have

826 610=2⋅5+4⋅52+2⋅53+2⋅54+4⋅55⁢2⋅56+2⋅58=22⋅131+48⋅1312=48⋅631+2⋅6312.

Therefore, by Lemma 5, Wn/2 contains Sylow p-subgroups for p=5,131,631. The collection

ℋ={X1,Wn/2,H2},

where H2 is a maximal subgroup containing Sylow 2-subgroups, satisfies condition (CI). By Theorem 1, we have η⁢(An)=3.

Example 1 also suggests the following:

Question C.

Are there infinitely many odd primes pi for which the subgroup Wn/2 of An, for n=2⁢p1⁢p2⁢…⁢pk, contains Sylow pi-subgroups for alli=1,…,k?

Equivalently,

Question C${}^{\prime}$.

Are there infinitely many odd primes pi for which the number 12⁢(nn/2), for n=2⁢p1⁢p2⁢…⁢pk, is coprime with every pi?

Note that a positive answer to the above questions will provide a different proof of Theorem 2.

3 Other finite simple groups

3.1 Classical groups of Lie type

We assume now that G is a finite simple classical group. We adopt the notation of [12] and, by [12, Proposition 2.9.1], we let G be one of the following groups:

PSLn⁢(q) for n≥2 and q≥7 when n=2,
PSpn⁢(q) for n≥2 even and q≥3,
PSUn⁢(q) for n≥3, q≥9 a square,
P⁢Ωn⁢(q) for n≥7 odd and q≥3 odd,
P⁢Ωn+⁢(q) for n≥4 even,
P⁢Ωn-⁢(q) for n≥4 even.

Proposition 3.

Assume that G is a finite simple classical group. Then Table 1 provides upper bounds for η⁢(G). In particular, we always have η⁢(G)≤4+2⁢ω⁢(n).

Proof.

The proof is based on a direct inspection of the indices of the maximal subgroups of the simple classical groups. The basic references are [12] when n≥13, and [1] otherwise. Here we limit our exposition to n≥13 and treat in detail only the case G=PSLn⁢(q); for the other groups, we just exhibit a list ℋ of subgroups having property (CI) and whose cardinality realizes the upper bound given in Table 1. When n≤12, better bounds (and, in some cases, explicit computations) can be found, according to the specific parameters of G.

Table 1

Upper bounds for η⁢(G) when G is a simple classical group.

G	η⁢(G)≤	d
PSLn⁢(q)	2+ω⁢(d)+ω⁢(n)	gcd⁡(n,q-1)
PSpn⁢(q)	4+ω⁢(n/2)
PSUn⁢(q)	4+ω⁢(d)+ω⁢(n)	gcd⁡(n,q+1)
P⁢Ωn⁢(q)	3
P⁢Ωn+⁢(q)	4+ω⁢(d)+ω⁢(n/2)	gcd⁡(4,qn-1)
P⁢Ωn-⁢(q)	4+ω⁢(n/2)

Case G=PSLn⁢(q). The only maximal subgroups of G containing Sylow p-subgroups, for p being the characteristic of the underlying field, are the maximal parabolic subgroups, namely, the conjugates of the various Pi, for i=1,…,n. We take P1 to be the stabilizer of a 1-dimensional subspace so that

|G:P1|=qn-1q-1.

Now, for every odd prime p dividing n, we set Rp to be a maximal subgroup of Aschbacher’s class 𝒞3 associated to the field extension 𝔽qp of 𝔽q. For the structure of Rp, as well as its order and properties, we refer the reader to [12, Proposition 4.3.6], (or [10, 19]). Such a group Rp contains Sylow t-subgroups for the various primes t such that p divides the order of q(modt), i.e., t dividing qp-1. Therefore, by taking all the Rp, for the primes p∣n, we can cover every prime dividing (qn-1)/(q-1) and not dividing q-1. In particular, we have

gcd(|G:P1|,|G:Rp|:p∣n)dividesgcd(qn-1q-1,q-1).

Note that every odd prime s, dividing both (qn-1)/(q-1) and q-1, is necessarily a divisor of n and so of d=gcd⁡(n,q-1). In general, a Sylow s-subgroup Qs is not contained in any of the Rp above (it does if and only if s2∤q-1) and, for this reason, we need to add at most ω⁢(d) more subgroups (the Sylow s-subgroups for s dividing d). Finally, if necessary, we need to add to our list (a maximal subgroup containing) a Sylow 2-subgroup Q2. Then

ℋ={P1,Rp,Qs,Q2:p∣n,s∣d}.

Case G=PSpn⁢(q). A list of subgroups having coprime indices is

ℋ={P1,A,B,Q2,Rp,Qs,Q2:p∣(n/2)},

where

P1 is the stabilizer of a point [12, Proposition 4.1.19],
A and B are given by
A≃{q-12.PGLn/2⁢(q)⁢.2∈𝒞2if⁢q⁢is odd [12, Proposition 4.2.5],On+⁢(q)∈𝒞8if⁢q⁢is even [12, Proposition 4.8.6],
B≃{On+⁢(q)∈𝒞8if⁢q,n/2⁢are even [12, Proposition 4.8.6],On-⁢(q)∈𝒞8if⁢q⁢is even and⁢n/2⁢is odd[12, Proposition 4.8.6],q+12.PGUn/2⁢(q)⁢.2∈𝒞3if⁢q⁢is odd [12, Proposition 4.3.7],
Q2 is a Sylow 2-subgroup,
Rp, for p∣(n/2), are maximal subgroups of class 𝒞3 [12, Proposition 4.3.10], each containing Sylow t-subgroups for primes t such that the order of q2(modt) is divisible by p.

Case G=PSUn⁢(q). The bound in Table 1 is reached by considering the following list of subgroups having coprime indices:

ℋ={P1,S1,W,Q2,Rp,Qs:2≠p∣n,s∣d},

where

P1 is the stabilizer of an isotropic point [12, Proposition 4.1.18],
S1 the stabilizer of a non-isotropic point [12, Proposition 4.1.4],
W≃PSpn⁢(q).(2,q-1)⁢(q+1,n/2)d∈𝒞5 when n is even [12, Proposition 4.5.6],
Q2 is a Sylow 2-subgroup and Qs a Sylow s-subgroup, for s∣d,
Rp, for odd primes p∣n, is a maximal subgroup of class 𝒞3 (see [12, Proposition 4.3.6]), each containing Sylow t-subgroups of G for the primes t such that the order of q(modt) is divisible by the prime p.

Case G=P⁢Ωn⁢(q), nq odd. A collection of three maximal subgroups having (CI) is

ℋ={P1,H+,H-},

where

P1 is the stabilizer of an isotropic point,
H+≃Ωn-1+⁢(q)⁢.2 is the stabilizer of a “plus” point [12, Proposition 4.1.16],
H-≃Ωn-1-⁢(q)⁢.2 is the stabilizer of a “minus” point [12, Proposition 4.1.16].

Case G=P⁢Ωn+⁢(q). We may take

ℋ={P1,H,U,R2,Rp,Q2,Qs:2≠p∣(n/2), 2≠s∣d},

where

P1∈𝒞1 is the stabilizer of an isotropic point [12, Proposition 4.1.20],
H is given by
H≃{Spn-2⁢(q)∈𝒞1if⁢q⁢is even [12, Proposition 4.1.7],Ωn-1⁢(q).a∈𝒞1if⁢q⁢is odd [12, Proposition 4.1.6],
where a∈{1,2},
U∈𝒞3 (see [12, Proposition 4.3.18]) containing Sylow t-subgroups for
2≠t∣(q+1) when 4∣n,
R2∈𝒞3 when n/2≥4 and n is even [12, Proposition 4.3.14],
Rp∈𝒞3, for 2≠p∣(n/2), when n/2 is not a prime [12, Proposition 4.3.14], otherwise Pn/2∈𝒞1 if n/2 is an odd prime [12, Proposition 4.1.20],
Q2 and Qs are respectively Sylow 2- and s-subgroups, for odd primes s∣d.

Case G=P⁢Ωn-⁢(q). Here we take

ℋ={P1,H,Rp,R,Q2:p∣(n/2)},

where

P1⁢𝒞1 is the stabilizer of an isotropic point,
H is given by
H≃{Ωn-1⁢(q)⁢.2∈𝒞1if⁢q⁢is even [12, Proposition 4.1.6],Ωn-1⁢(q).a∈𝒞1if⁢q⁢is odd [12, Proposition 4.1.6],
where a∈{1,2},
R≃q+1(q+1,4).Ur⁢(q).[(q+1,r)]∈𝒞3 when n/2 is odd [12, Proposition 4.2.18],
Rp, for primes p∣n/2, are maximal subgroups of class 𝒞3 (see [12, Proposition 4.2.16]) containing Sylow t-subgroups of G for those odd primes t such that the order of q(modt) is divisible by p,
Q2 a Sylow 2-subgroup. ∎

We conclude the case of classical groups by noting that, contrary to what happens for alternating groups (Theorem 1), there exist classical simple groups having indices covering number two, without having maximal subgroups of prime power index. One example is the group PSL4⁢(5).

3.2 Groups of exceptional type

Let G be an exceptional simple group, namely, one of the following groups:

B22⁢(q) with⁢q=22⁢m+1≥8,

G2⁢(q) with⁢q≥3,G22⁢(q) with⁢q=32⁢m+1≥27,

F4⁢(q) with⁢q≥3,F42⁢(q) with⁢q=22⁢m+1≥8,

D43⁢(q), E6⁢(q), E62⁢(q), E7⁢(q) and E8⁢(q). Note that we need not treat B22⁢(2) (which is solvable), G2⁢(2)≃PSU3⁢(3)⁢.2, G22⁢(3)≃PSL2⁢(8)⁢.3 and F42⁢(2)′, which will be treated in Section 3.3.

Proposition 4.

For every finite exceptional simple group of Lie type G, we have η⁢(G)≤20.

Proof.

Let p be a prime dividing |G|. If p is different from the characteristic of G and p does not divide 2⁢|W|, where W is the Weyl subgroup associated to G, then every Sylow p-subgroup of G lies in a Sylow d-torus, for a unique d, for which the cyclotomic value Φd⁢(q) divides |G| (see [19, Theorem 25.14]). Since exceptional groups have bounded rank, the number of such d is bounded and therefore so are the values of η⁢(G).

Table 2 provides reasonably good bounds for η⁢(G) when G is one of the simple groups of exceptional type. These results have been obtained by analyzing the indices of the maximal subgroups of G; these have been classified in the papers [22, 13, 14, 18, 17]. ∎

Table 2

Upper bounds for η⁢(G) when G is an exceptional group.

G	η⁢(G)≤	G	η⁢(G)≤
B22⁢(q)	3	D43⁢(q)	5
G2⁢(q)	3	E6⁢(q)	7
G22⁢(q)	4	E62⁢(q)	6
F4⁢(q)	4	E7⁢(q)	6
F42⁢(q)	6	E8⁢(q)	14

3.3 The case of sporadic groups

Table 3 provides the exact values of η⁢(G), for (almost all) the sporadic simple groups G and the Tits group F42⁢(2)′. Indeed, for the groups Fi24′, B and M, we furnish an upper bound (which we think is probably the exact value of η⁢(G)), as the list of maximal subgroups of these groups is still incomplete. These results have been obtained by a direct inspection in [6] of the maximal subgroups of G.

Table 3

A comparison between η⁢(G) and ω⁢(|G|) for sporadic groups and the Tits group.

G	η⁢(G)	ω⁢(\|G\|)	G	η⁢(G)	ω⁢(\|G\|)	G	η⁢(G)	ω⁢(\|G\|)
M11	2	4	Co3	4	6	B	≤5	11
M12	3	4	Co2	3	6	M	≤10	15
M22	3	5	Co1	5	7	J1	3	6
M23	2	6	He	3	5	O⁢’⁢N	5	7
M24	2	6	Fi22	3	6	J3	4	5
J2	3	4	Fi23	4	8	Ly	4	8
Suz	3	6	Fi24′	≤5	9	Ru	4	6
HS	3	5	HN	5	6	J4	5	10
McL	3	5	Th	5	7	F42⁢(2)′	3	4

4 Normal primary coverings and the function γ0⁢(G)

Given a finite group G, let G0 be the set of its primary elements, namely,

G0={g∈G∣|g|⁢is a prime power}.

Definition 1.

A normal primary covering for G is a collection of conjugacy classes of subgroups of G whose union contains G0.

If G is a finite group which is not a cyclic p-group, we define γ0⁢(G) to be the cardinality of a smallest normal primary covering for G, i.e., the smallest natural integer such that

G0⊆⋃i=1γ0⁢(G)⋃g∈GHig,

for some proper (pairwise non-conjugate) subgroups Hi of G.

This definition is suggested by the analogous function γ⁢(G) (defined in [3]) as the smallest positive number of conjugacy classes of subgroups that covers all of G.

The following lemma collects some basic facts about the function γ0.

Proposition 5.

Let G be a finite group. Then the following holds.

If G is a non-cyclic p-group, then γ0⁢(G)=γ⁢(G)=p+1.
If G is not a p-group, then 2≤γ0⁢(G)≤η⁢(G).

Proof.

(1) This is trivial from the definitions of the functions γ0 and γ.

(2) The fact that γ0⁢(G)≤η⁢(G) is immediate from the definitions of the two functions. The fact that γ0⁢(G)≥2 is [9, Theorem 1], and it depends on CFSG. ∎

Lemma 7.

Let p be a prime divisor of n. Then any p-element of An acting fixed-point-freely on {1,2,…,n} lies in a conjugate of the subgroup Wn/p.

Proof.

Let p be odd (the proof for p=2 is just a slight modification). Write n=r⁢p, and let τ be a p-element that has no fixed points. Assume that τ is the disjoint product of cycles ci, for i=1,2,…,h, and that each ci has order pαi so that it can be written as ci=(a1i,a2i,…,apαii). Since τ acts fixed-point-freely, we necessarily have n=∑i=1hpαi, thus r=∑i=1hpαi-1. Now, for every j=0,1,…,p-1, we set

Δj:={asi∣i=1,…,h,s≡j(modp)}.

Note that |Δj|=∑i=ihpαi-1=r for every j and that τ stabilizes the partition {Δ0,Δ1,…,Δp-1}. Since An acts transitively on the set of partitions of r-size blocks, τ lies in a conjugate of Wr. ∎

Proposition 6.

If n≥5 and n is either pk or pr⁢q, with p≠q, then γ0⁢(An)=2. In particular, γ0 and η are different arithmetic functions.

Proof.

The case n=pk is trivial by Proposition 5 (2) and Theorem 1.

Let n=pr⁢q. We construct a normal primary covering of cardinality 2 by taking the subgroups X1 and Wpr. Indeed, any given primary element of An with no fixed point is either a p-element or a q-element. In any case, it lies in a conjugate of Wpr by Lemma 5 and Lemma 7.

When p and q are different primes we have γ0⁢(Apr⁢q)=2 and η⁢(Apr⁢q)=3. ∎

Communicated by Andrea Lucchini

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Received: 2018-11-19

Revised: 2019-03-14

Published Online: 2019-05-07

Published in Print: 2019-11-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2018-0212