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(H) is at most d(G) + 4.