## Abstract

Let G be a finite group and let d(G) be the minimal number of generators

for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We

prove that d(H) 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.

We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.

We then apply our results to the study of permutation groups. In particular we

show that if G is a finite primitive permutation group with point stabilizer H, then

d(H) is at most d(G) + 4.

for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We

prove that d(H) 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.

We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.

We then apply our results to the study of permutation groups. In particular we

show that if G is a finite primitive permutation group with point stabilizer H, then

d(H) is at most d(G) + 4.

Original language | English |
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Pages (from-to) | 59-95 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 248 |

DOIs | |

Publication status | Published - 25 Nov 2013 |