GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES
Open Access
- 7 November 2017
- journal article
- research article
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
Abstract
Let be a finite almost simple group. It is well known that can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of . 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 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 for which there is a prime power such that is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.This publication has 27 references indexed in Scilit:
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