Generation of second maximal subgroups and the existence of special primes

Tim Burness, Martin Liebeck, Aner Shalev

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
265 Downloads (Pure)

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)
Original languageEnglish
Article numbere25
Number of pages41
JournalForum of Mathematics, Sigma
Volume5
Early online date7 Nov 2017
DOIs
Publication statusE-pub ahead of print - 7 Nov 2017

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