# 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 $\left| S/\mathrm{Z}(S)\right| > C$, $\beta_{j}$ $(j=1,\ldots,M)$ are any automorphisms of $S$, and $q_{j}$ $(j=1,\ldots,M)$ 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.

## Authors

Nikolay Nikolov

New College, Oxford University, Oxford OX1 3BN, United Kingdom

Dan Segal

All Souls College, Oxford University, Oxford OX1 4AL, United Kingdom