On finitely generated profinite groups, II: Products in quasisimple groups

Abstract

We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group 5, given 2D arbitrary automorphisms of 5, every element of 5 is equal to a product of D 'twisted commutators' defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if 5 is a finite quasisimple group with S/Z(S)\ > C, βj (j = 1,... , M) are any automorphisms of 5, and qj (j = 1,... , M) are any divisors of q, then there exist inner automorphisms αj of S such that S = Π1M[S, (αjβj)qj]. These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.