## 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( G) - 1 ≤ d( H) ≤ d( G) + 4.

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

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 248 |

DOIs | |

State | Published - 5 Nov 2013 |

### Bibliographical note

Funding Information:The first author acknowledges the support of EPSRC grant EP/I019545/1 . The second and third authors acknowledge the support of EPSRC grant EP/H018891/1 . The third author also acknowledges the support of an Advanced ERC Grant, an Israel Science Foundation Grant, and the Miriam and Julius Vinik Chair in Mathematics which he holds.

## Keywords

- Finite simple groups
- Maximal subgroups
- Minimal generation
- Primitive permutation groups