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.
Bibliographical noteFunding Information:
The authors are grateful for the hospitality of the Centre Interfacultaire Bernoulli at EPFL, where this work was completed. The second and third authors acknowledge the support of EPSRC Mathematics Platform grant EP/I019111/1. The third author also acknowledges the support of Advanced ERC Grant 247034, an ISF grant 1117/13, and the Miriam and Julius Vinik Chair in Mathematics which he holds.
© 2017 The Author(s).