Generation and random generation: From simple groups to maximal subgroups

Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.