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


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.


Nikolay Nikolov

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

Dan Segal

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