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On the Number of Maximal Subgroups in a Finite Group J. Group Theory 18 (2015), 535–551 DOI 10.1515/jgth-2015-0001 © de Gruyter 2015 On the number of maximal subgroups in a finite group L.-K. Lauderdale Communicated by Nigel Boston Abstract. For a finite group G, let m.G/ denote the set of maximal subgroups of G and .G/ denote the set of primes which divide G . In [4], it is proven that m.G/ .G/ when G is cyclic and if G is noncyclic m.G/j j .G/ p, wherej p .G/j D j is thej smallest prime that divides G . In thisj paper,j we j producej C two new lower2 bounds for m.G/ , both of which considerj j all of the primes in .G/. j j 1 Introduction For any finite group G, let .G/ denote the set of primes which divide G and let j j m.G/ denote the set of maximal subgroups of G. Lower bounds for the number of maximal subgroups have previously been investigated. In [4], three lower bounds are proven. In general, m.G/ .G/ and equality holds if and only if G is j j j j cyclic. Under further assumptions on the structure of G, this lower bound was improved. If G is noncyclic, then m.G/ .G/ p, where p .G/ is the j j j j C 2 smallest prime that divides G . Additionally, if G has a noncyclic Sylow subgroup j j and q .G/ is the smallest prime such that Q Syl .G/ is noncyclic, then it is 2 2 q proven that m.G/ .G/ q: (1.1) j j j j C Observe that given any set of distinct primes q1; q2; : : : ; qn where q1 is the ¹ º smallest, there exists a finite group G with .G/ q1; q2; : : : ; qn and m.G/ .G/ q1: D ¹ º j j D j j C For example, G Zq1 Zq1q2 qn achieves the lower bound in (1.1). However, D the previously stated lower bounds can be restrictive because they consider at most one prime in .G/. For instance, let G S10 Z11 Z143, where S10 D is the symmetric group on ten letters. Then .G/ 6 and G has a noncyclic j j D Sylow 2-subgroup. Therefore q 2 and equation (1.1) yields m.G/ 8, how- D j j ever m.G/ 4002. The lower bound in (1.1) can be drastically improved with j j D minimal assumptions on the structure of the group. 536 L.-K. Lauderdale In the present paper, we improve the existing lower bound for m.G/ by par- j j titioning .G/ into three sets. We consider the set of primes p for which G is not p-solvable, denoted .G/, and the set of primes q for which G is q-solvable. We further subdivide the latter set of primes into the set of primes which corre- spond to cyclic Sylow subgroups of G, denoted .G/, and the set of primes which correspond to noncyclic Sylow subgroups of G, denoted .G/. Under these con- siderations we obtain two new lower bounds for m.G/ . In addition to the sets j j .G/, .G/ and .G/, the first lower bound includes a further assumption on G. We assume that the nonabelian composition factors of G satisfy a technical prop- erty, namely Property B, which is defined in Definition 2.5. Theorem 1.1. Let G be a finite group. If the nonabelian composition factors of G satisfy Property B, then X X m.G/ .G/ p p: (1.2) j j j j C C p Ä.G/ p .G/ 2 2 Property B is not a very restrictive property and we conclude this paper by pro- viding some examples of simple groups that satisfy Property B (see Section 8). However, not all finite simple groups satisfy Property B and knowing the compo- sition factors of a group may increase the possible lower bound. For a group G, we introduce below the invariant ˛.G/ which takes into account more finely the influence of the composition factors on our problem. The following theorem does not require any special condition on the composition factors. Theorem 1.2. If G is a finite group, then X X m.G/ .G/ .p 1/ ˛.G/ .p 1/: (1.3) j j j j C C C C p Ä.G/ p .G/ 2 2 Notice that Theorem 1.2 implies Theorem 1.1 when ˛.G/ 1. The invariant ˛.G/ is closely connected to the values ˛.C/ as C runs over the nonabelian com- position factors of G. In the penultimate section (see Section 7), we analyze ˛.G/ and see that ˛.C/ tends to infinity for many families of finite simple groups C. Although, we provide many examples of finite simple groups which satisfy both ˛.C/ > 1 and Property B, our results do not use the classification of finite simple groups. Both Theorem 1.1 and Theorem 1.2 improve the lower bound in equation (1.1). Reconsidering the group G S10 Z11 Z143, the lower bound in (1.3) yields D m.G/ 4001, a vast improvement over the lower bound in (1.1). In most cases j j Theorem 1.2 gives a stronger result than Theorem 1.1. However, it is possible On the number of maximal subgroups in a finite group 537 for the lower bound in (1.2) to be sharper than the lower bound in (1.3) because not all finite groups G satisfy ˛.G/ 1. For instance, ˛.A6/ < 1 where A6 is the alternating group on six letters, and ˛.PSL.2; 7// < 1. It can be shown that both A6 and PSL.2; 7/ satisfy Property B and thus Theorem 1.1 is applicable. Theorem 1.1 gives the sharper bound for G PSL.2; 7/ Z11 Z143. In this D case, the lower bound in (1.3) yields m.G/ 21, but the lower bound in (1.2) j j yields m.G/ 28. j j 2 Definitions To improve the existing lower bounds for m.G/ , we first partition .G/ into three j j sets: .G/, .G/ and .G/. Definition 2.1. Let G be a finite group. Define .G/ as the set of p .G/ such that G is p-solvable and P Syl .G/ 2 2 p is cyclic. Define .G/ as the set of p .G/ such that G is p-solvable and P Syl .G/ 2 2 p is noncyclic. Define .G/ as the set of p .G/ such that p divides the order of a nonabel- 2 ian chief factor of G. To aid in counting some of the maximal subgroups of G with index a .G/-num- ber, we make the following definitions: Definition 2.2. Let S be a nonabelian simple group and assume M is a maximal subgroup of S. We say that M satisfies Property A if N .M /S Aut.S/. Let Aut.S/ D mA.S/ denote the set of maximal subgroups of S that satisfy Property A. Definition 2.3. Let G be a finite group and suppose C is a nonabelian composition P P factor of G. Define .C/ p .C/.p 1/ and .G/ p .G/.p 1/. D 2 C D 2 C Definition 2.4. Let G be a finite group and suppose C is a nonabelian composition mA.C/ factor of G. Set ˛.C/ j j . If G is not solvable, set D .C/ ˛.G/ min ˛.C/ C is a nonabelian composition factor of G D ¹ W º and if G is solvable, then set ˛.G/ 0. D Definition 2.5. Let S be a nonabelian finite simple group. We say that S satisfies Property B if m.G/ .S/, for all subgroups G such that S G Aut.S/. j j Ä Ä 538 L.-K. Lauderdale 3 Preliminary results In this section, we collect a few results which will aid in the proof of the main theorems. Lemma 3.1. Let p be a prime number and suppose that P is a noncyclic p-group. Let P=ˆ.P / be the elementary abelian group of rank k, where k 2 and ˆ.P / denotes the Frattini subgroup of P . Then pk 1 m.P / : j j D p 1 In particular, m.P / p 1. j j C Proof. Since P=ˆ.P / is an elementary abelian group of rank k, we have pk 1 m.P=ˆ.P // : j j D p 1 Because ˆ.P / is contained in every maximal subgroup of P , pk 1 m.P / m.P=ˆ.P // : j j D j j D p 1 pk 1 Observe that p 1 is smallest when k 2. In this case, D pk 1 p 1; p 1 D C so that m.P / p 1. j j C Theorem 3.2 (Maschke). Let G be a finite group and let F be a field whose char- acteristic does not divide G . If V is any FG-module and U is any submodule j j of V , then V has a submodule W such that V U W . D ˚ Proof. See for example [2, Section 18.1, Theorem 1]. Theorem 3.3. For a finite cyclic group G, m.G/ .G/ . j j D j j Proof. See for example [4, Theorem 3.1]. Lemma 3.4. Suppose that G is a finite group, and let .G/ p1; p2; : : : ; pn . D ¹ º For each i 1; 2; : : : ; n , let Pi Syl .G/. If G P1 P2 Pn, then 2 ¹ º 2 pi D m.G/ m.P1/ m.P2/ m.Pn/ .
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