## Abstract

Let G be a finite almost simple group. It is well known that G can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of G. In this paper we consider subgroups at the next level of the subgroup lattice – the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of G is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes r for which there is a prime power q such that (q r − 1)/(q − 1) is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

2010 Mathematics Subject Classification: 20D06 (primary); 20D30, 20P05 (secondary)

2010 Mathematics Subject Classification: 20D06 (primary); 20D30, 20P05 (secondary)

Original language | English |
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Article number | e25 |

Number of pages | 41 |

Journal | Forum of Mathematics, Sigma |

Volume | 5 |

Early online date | 7 Nov 2017 |

DOIs | |

Publication status | E-pub ahead of print - 7 Nov 2017 |