The author proves in this paper that every profinite group G with polynomial subgroup growth is boundedly generated; that is, it is a product of finitely many procyclic subgroups. This answers a question of P. Zalesskii. By contrast, if G is a boundedly generated group, then the subgroup growth of G is at most nclogn . As a byproduct, a short, elementary proof demonstrates that Aut(Fr) (for r [ges ] 2) and many other related groups are not boundedly generated.

Let A be a group of automorphisms of the finite group G such that ([mid ]A[mid ], [mid ]G[mid ])=1. Then [mid ]A[mid ]<[mid ]G[mid ]2, and the exponent 2 here is best possible. If, moreover, A is nilpotent of class at most 2, then [mid ]A[mid ]<[mid ]G[mid ]. If A is abelian, then A has a regular orbit on G.