TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 430870 10.1155/2013/430870 430870 Research Article On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups Shi Jiangtao 1 Zhang Cui 2 Hoff da Silva J. Park J. 1 School of Mathematics and Information Science Yantai University Yantai 264005 China ytu.edu.cn 2 Department of Applied Mathematics and Computer Science Technical University of Denmark 2800 Lyngby Denmark dtu.dk 2013 27 10 2013 2013 03 08 2013 11 09 2013 2013 Copyright © 2013 Jiangtao Shi and Cui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)|2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if GPSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).

1. Introduction

In this paper, all groups are assumed to be finite. It seems interesting to investigate the influence of some arithmetic properties of noncyclic proper subgroups on the solvability of groups. In , Li and Zhao proved that any group having at most three conjugacy classes of noncyclic proper subgroups is solvable, and a group G having exactly four conjugacy classes of noncyclic proper subgroups is nonsolvable if and only if GPSL(2,5) or SL(2,5). As a generalization of the above result, we showed that any group having at most three conjugacy classes of nonnormal noncyclic proper subgroups is solvable, and a group G having exactly four conjugacy classes of nonnormal noncyclic proper subgroups is nonsolvable if and only if GPSL(2,5) or SL(2,5) (see ).

Let G be a group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. It is clear that a group G with 𝒩𝒞(G)= is either a cyclic group or a minimal noncyclic group, and a group G with 𝒩𝒞(G)={1} is a group in which every noncyclic proper subgroup is normal. In , we also obtained a complete classification of groups G in which every noncyclic proper subgroup is nonnormal; all such groups G satisfy 1𝒩𝒞(G).

By |𝒩𝒞(G)| we denote the order of 𝒩𝒞(G). Note that we cannot ensure that 1𝒩𝒞(G) for any solvable group G with |𝒩𝒞(G)|=n1. For example, let GD2pn be a dihedral group of order 2pn, where n1 and p is an odd prime. Then 𝒩𝒞(D2pn)={p,p2,,pn}, so 1𝒩𝒞(D2pn). For the nonsolvable group of the smallest order PSL(2,5), it is easy to see that 𝒩𝒞(PSL(2,5))={5,6,10}, and so |𝒩𝒞(PSL(2,5))|=3.

For the influence of |𝒩𝒞(G)| on the solvability of groups, we have the following result, the proof of which is given in Section 3.

Theorem 1.

Let G be a group.

If |𝒩𝒞(G)|2, then G is solvable.

G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if GPSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).

The following two corollaries are direct consequences of Theorem 1.

Corollary 2.

Let G be a group with |𝒩𝒞(G)|3. Then G is nonsolvable if and only if 𝒩𝒞(G)={5,6,10} or {14,78,91}.

Corollary 3.

Let G be a group and 𝒩𝒯(G) the set of the numbers of conjugates of nontrivial subgroups of G.

If |𝒩𝒯(G)|2, then G is solvable.

G is a nonsolvable group with |𝒩𝒯(G)|=3 if and only if GPSL(2,13).

Let G be a group and 𝒩𝒞*(G) the set of the numbers of conjugates of nonnormal noncyclic proper subgroups of G. Obviously 𝒩𝒞*(G)𝒩𝒞(G).

Arguing as in the proof of Theorem 1, we can obtain the following result.

Theorem 4.

Let G be a group. If |𝒩𝒞*(G)|2, then G is solvable.

Remark 5.

If we assume that G is a nonsolvable group with |𝒩𝒞*(G)|=3, we cannot get that Φ(G)=Z(G). For example, let GPSL(2,5)×p, where p7 is a prime. It is easy to see that |𝒩𝒞*(G)|=3. But Φ(G)=1 and Z(G)=p.

Let G be a group and 𝒩𝒜(G) the set of the numbers of conjugates of nonabelian proper subgroups of G. Obviously 𝒩𝒜(G)𝒩𝒞(G). Arguing as in the proof of Theorem 1, we can also obtain the following result.

Theorem 6.

Let G be a group. If |𝒩𝒜(G)|2, then G is solvable.

2. Preliminaries

In this section, we collect some essential lemmas needed in the sequel.

Lemma 7 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let G be a group. If all nonnormal maximal subgroups of G have the same order, then G is solvable.

Lemma 8 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let G be a nonsolvable group having exactly two classes of nonnormal maximal subgroups of the same order; then G/S(G)PSL(2,7), where S(G) is the largest solvable normal subgroup of G.

Lemma 9 (see [<xref ref-type="bibr" rid="B6">5</xref>, <xref ref-type="bibr" rid="B8">6</xref>]).

Let G be a group having exactly n classes of maximal subgroups of the same order, where 1n3; then one of the following statements holds:

suppose that G is a group with n=1, and then G is a p-group for some prime p;

suppose that G is a nonsolvable group with n=2, and then G/Φ(G)(23iPSL(2,7))×7j, where i,j=0,1,, and 23iPSL(2,7) is a semidirect product of the normal subgroup 23i and the subgroup PSL(2,7);

suppose that G is a nonsolvable group with n=3, and then G/S(G)A6; PSL(2,q), q=11,13,23,59,61; PSL(3,3); U3(3);PSL(5,2); PSL(2,2f), and   f is a prime; PSL(2,7)×PSL(2,7)××PSL(2,7).

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

The proof of Theorem 1 follows from the following two lemmas.

Lemma 10.

Let G be a group. If |𝒩𝒞(G)|2, then G is solvable.

Proof.

Assume that G is nonsolvable. Then by [7, Exercise 10.5.7], all maximal subgroups of G are noncyclic. Let 𝒮(G) be the set of the numbers of conjugates of maximal subgroups of G. It follows that 𝒮(G)𝒩𝒞(G). Then |𝒮(G)|2.

Suppose that 1𝒮(G). Since G is nonsolvable, G must have nonnormal maximal subgroups. Let M be any nonnormal maximal subgroup of G; one has |G:NG(M)|=|G:M|. Since |𝒮(G)|2, we know that G has at most one class of nonnormal maximal subgroups of the same order. It follows that G is solvable by Lemma 7, a contradiction.

Suppose that 1𝒮(G). It follows that all maximal subgroups of G are nonnormal. By the hypothesis, G has at most two classes of maximal subgroups of the same order. Since G is nonsolvable and G has no normal maximal subgroups, one has G/Φ(G)23iPSL(2,7) by Lemma 9 (1) and (2), where i=0,1,. It is easy to see that 𝒩𝒞(G/Φ(G))𝒩𝒞(G) and |𝒩𝒞(23iPSL(2,7))|>2. It follows that |𝒩𝒞(G)|>2, a contradiction.

Thus, our assumption is not true, so G is solvable.

Lemma 11.

A group G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if GPSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).

Proof.

The sufficiency part is evident, and we only need to prove the necessity part.

By the hypothesis, |𝒮(G)|3. We claim that (1)1𝒮(G).

Otherwise, assume that 1𝒮(G). Then G has at most two classes of nonnormal maximal subgroups of the same order. Since G is nonsolvable, one has G/S(G)PSL(2,7) by Lemmas 7 and 8. It is easy to see that 𝒩𝒞(G/S(G))𝒩𝒞(G) and |𝒩𝒞(PSL(2,7))|>3. It follows that |𝒩𝒞(G)|>3, a contradiction. Thus, 1𝒮(G).

Since |𝒮(G)|3, we have that G has at most three classes of maximal subgroups of the same order.

By Lemma 9 (1), G cannot have exactly one class of maximal subgroups of the same order.

If G has exactly two classes of maximal subgroups of the same order, according to Lemma 9 (2), one has G/Φ(G)23iPSL(2,7) since G has no normal maximal subgroups, where i=0,1,. Since |𝒩𝒞(23iPSL(2,7))|>3, it follows that |𝒩𝒞(G)|>3, a contradiction.

Thus, G has exactly three classes of maximal subgroups of the same order. By Lemma 9 (3), G/S(G) might be isomorphic to A6 or PSL(2,q),q=11,13,23,59,61 or PSL(3,3) or U3(3) or PSL(5,2) or PSL(2,2f), and f is a prime or PSL(2,7)×PSL(2,7)××PSL(2,7). If G/S(G) is an isomorphism to A6 or PSL(2,q), q=11,23,59,61 or PSL(3,3) or U3(3) or PSL(5,2) or PSL(2,2f), and   f is an odd prime or PSL(2,7)×PSL(2,7)××PSL(2,7). It is easy to see that |𝒩𝒞(G/S(G))|>3 by [8, 9], which implies that |𝒩𝒞(G)|>3, a contradiction. Thus, G/S(G)PSL(2,4)PSL(2,5) or PSL(2,13).

Note that 1𝒮(G) and |𝒮(G)|=|𝒩𝒞(G)|=3. It follows that 1𝒩𝒞(G), so S(G) is cyclic. We claim that (2)Φ(G)=S(G).

Otherwise, assume that Φ(G)<S(G). Let M be a maximal subgroup of G such that S(G)M. Then G=S(G)M. It is obvious that S(G)MM. Moreover, S(G)MS(G), since S(G) is cyclic. It follows that S(G)MG. Therefore, G/(S(G)M)=S(G)/(S(G)M)M/(S(G)M). Let G-=G/(S(G)M),S-(G)=S(G)/(S(G)M), and M-=M/(S(G)M). By N/C-theorem, NG-(S-(G))/CG-(S-(G))Aut(S-(G)). That is, G-/CG-(S-(G))=S-(G)M-/CG-(S-(G)))Aut(S-(G)). Note that Aut(S-(G)) is abelian since S-(G) is cyclic. Moreover, M-S(G)M/S(G)=G/S(G) is a nonabelian simple group and S-(G)M-/CG-(S-(G))(S-(G)M-/S-(G))/(CG-(S-(G))/S-(G)). Here S-(G)M-/S-(G)M-. Therefore, one has CG-(S-(G))/S-(G)=1 or CG-(S-(G))/S-(G)=S-(G)M-/S-(G)=G-/S-(G). If CG-(S-(G))/S-(G)=1, it follows that S-(G)M-/S-(G)Aut(S-(G)) is abelian, a contradiction. If CG-(S-(G))/S-(G)=G-/S-(G), then S-(G)Z(G-). It follows that G-=S-(G)×M- and then MG; this contradicts that all maximal subgroups of G are nonnormal. Thus, our assumption is not true, so Φ(G)=S(G).

It follows that G/Φ(G)PSL(2,5) or PSL(2,13).

If Φ(G)=1, then GPSL(2,5) or PSL(2,13).

Next, suppose that Φ(G)1. Let p be any prime divisor of |Φ(G)|. We claim that p2. Otherwise, assume that p>2. Let T be a subgroup of Φ(G) such that Φ(G)/Tp. That is, Φ(G/T)p. Then (G/T)/p(G/T)/Φ(G/T)=(G/T)/(Φ(G)/T)G/Φ(G)PSL(2,5) or PSL(2,13). Since p>2 and Schur multipliers of both PSL(2,5) and PSL(2,13) are 2, we have that G/TPSL(2,5)×p or PSL(2,13)×p. Note that |𝒩𝒞(PSL(2,5)×p)|>3 and |𝒩𝒞(PSL(2,13)×p)|>3. It follows that |𝒩𝒞(G)|>3, a contradiction. Thus, p2, so Φ(G) is a cyclic 2-group. If |Φ(G)|=2n>2, let L be a subgroup of Φ(G) such that Φ(G)/L2. Then (G/L)/2(G/L)/Φ(G/L)=(G/L)/(Φ(G)/L)G/Φ(G)PSL(2,5) or PSL(2,13). We have that G/LSL(2,5) or SL(2,13). Let M be a subgroup of L such that L/M2. Then (G/M)/2(G/M)/(L/M)G/LSL(2,5) or SL(2,13). Since Schur multipliers of both SL(2,5) and SL(2,13) are trivial, we have that G/MSL(2,5)×2 or SL(2,13)×2; this contradicts that all maximal subgroups of G are nonnormal. Thus, |Φ(G)|=2. It follows that GSL(2,5) or SL(2,13).

Lemmas 10 and 11 combined together give Theorem 1.

Acknowledgments

The authors are grateful to the referees who gave valuable comments and suggestions. Jiangtao Shi was supported by NSFC (Grant nos. 11201401 and 11361075). Cui Zhang was supported by H.C. Ørsted Postdoctoral Fellowship at DTU (Technical University of Denmark).

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