Maximal subgroups in composite finite groups.

*(English)*Zbl 0591.20020The purpose of the paper is to present a method for translating the problem of finding all maximal subgroups of finite groups into questions concerning groups that are nearly simple, i.e. groups that have only one maximal normal subgroup and that is nonabelian and simple.

Let G be a finite group and M is a minimal normal subgroup of G. Obviously in view of an inductive approach we are interested only in maximal subgroups of G not containing M. The following three cases take place:

1. M is abelian. In this case the maximal subgroups of G not containing M one can describe in terms of the first cohomology group \(H^ 1(G/M,M).\)

2. M is nonabelian and nonsimple. This case is the principal part of the paper. Here the following result is obtained (Theorem 4.3): Let \(K\triangleleft M\), \(N=N_ G(K)\) and \(M=\prod_{t\in G mod N}M/t^{- 1}Kt\). Then one can describe the set of the conjugacy classes of the maximal subgroups of G not containing M in terms of the factors G/M, N/K, and M/K.

3. M is nonabelian simple. In this case we are interested only in corefree maximal subgroups of G, i.e. subgroups which contain no nontrivial normal subgroups of G. Here the following result is obtained (Corollary 5.2): Let G be a not nearly simple group. Then G has no corefree maximal subgroups unless there are precisely two minimal normal subgroups, say \(M_ 1\) and \(M_ 2\), and the factor groups \(G/M_ 1\), \(G/M_ 2\) are isomorphic nearly simple groups.

Let G be a finite group and M is a minimal normal subgroup of G. Obviously in view of an inductive approach we are interested only in maximal subgroups of G not containing M. The following three cases take place:

1. M is abelian. In this case the maximal subgroups of G not containing M one can describe in terms of the first cohomology group \(H^ 1(G/M,M).\)

2. M is nonabelian and nonsimple. This case is the principal part of the paper. Here the following result is obtained (Theorem 4.3): Let \(K\triangleleft M\), \(N=N_ G(K)\) and \(M=\prod_{t\in G mod N}M/t^{- 1}Kt\). Then one can describe the set of the conjugacy classes of the maximal subgroups of G not containing M in terms of the factors G/M, N/K, and M/K.

3. M is nonabelian simple. In this case we are interested only in corefree maximal subgroups of G, i.e. subgroups which contain no nontrivial normal subgroups of G. Here the following result is obtained (Corollary 5.2): Let G be a not nearly simple group. Then G has no corefree maximal subgroups unless there are precisely two minimal normal subgroups, say \(M_ 1\) and \(M_ 2\), and the factor groups \(G/M_ 1\), \(G/M_ 2\) are isomorphic nearly simple groups.

Reviewer: E.Komissartschik

##### MSC:

20D05 | Finite simple groups and their classification |

20E28 | Maximal subgroups |

20D30 | Series and lattices of subgroups |

##### Keywords:

maximal subgroups of finite groups; maximal normal subgroup; minimal normal subgroup; first cohomology group; corefree maximal subgroups; nearly simple groups
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##### References:

[1] | Aschbacher, M; Scott, L, Maximal subgroups of finite groups, J. algebra, 92, 44-80, (1985) · Zbl 0549.20011 |

[2] | Cameron, P.J, Finite permutation groups and finite simple groups, Bull. London math. soc., 13, 1-22, (1981) · Zbl 0463.20003 |

[3] | Gross, F; Kovács, L.G, On normal subgroups of groups which are direct products, J. algebra, 90, 133-168, (1984) · Zbl 0594.20018 |

[4] | Lane, S.Mac, Homology, (), 114 |

[5] | Robinson, Derek J.S, A course in the theory of groups, () · Zbl 0836.20001 |

[6] | Scott, Leonard L, Representations in characteristic p, (), 319-331 |

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