Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4665954 | Advances in Mathematics | 2013 | 37 Pages |
Let G be a finite group and let d(G)d(G) be the minimal number of generators for G . It is well known that d(G)=2d(G)=2 for all (non-abelian) finite simple groups. We prove that d(H)⩽4d(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)+4d(G)−1⩽d(H)⩽d(G)+4.