Generation of second maximal subgroups and the existence of special primes

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Title: Generation of second maximal subgroups and the existence of special primes
Authors: Liebeck, MW
Burness, TC
Shalev, A
Item Type: Journal Article
Abstract: Let G be a finite almost simple group. It is well known that G can be generated by three 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.
Issue Date: 7-Nov-2017
Date of Acceptance: 13-Aug-2017
ISSN: 2050-5094
Publisher: Cambridge University Press (CUP)
Journal / Book Title: Forum of Mathematics, Sigma
Volume: 5
Copyright Statement: © The Author(s) 2017 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Publication Status: Published
Article Number: e25
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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