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 $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ 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 $S$ is a finite quasisimple group with $|S/\mathrm{Z}(S)|>C$, ${\beta }_j$ $(j=1,...,M)$ are any automorphisms of $S$, and $q_j$ $(j=1,...,M)$ are any divisors of $q$, then there exist inner automorphisms ${\alpha }_j$ of $S$ such that $S={\prod }_1^M[S,({\alpha }_j{\beta }_j{)}^{q_j}]$. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.

34 pages