Generation of second maximal subgroups and the existence of special primes

TL;DR

This paper investigates the generation properties of second maximal subgroups in finite almost simple groups, establishing bounds on generators and linking these properties to a deep open problem in Number Theory.

Contribution

It proves that second maximal subgroups are generated by a bounded number of elements, except possibly for some Lie type groups, and relates this to the finiteness of certain primes in number theory.

Findings

Bound on generators for most second maximal subgroups

Connection between subgroup generation bounds and prime number problems

Applications to random generation and polynomial growth

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.