Generation of second maximal subgroups and the existence of special primes

2016/11/18 by Timothy C. Burness, Martin W. Liebeck, Burness, Timothy C. +3
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1611.06196

30 pages

arxiv created 2016/11/18 · openalex publication_date 2016/11/18 · arxiv updated 2016/11/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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.

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