On finitely generated profinite groups II, products in quasisimple groups

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

2006-04-18

arxiv.org/abs/math/0604400v1

Mathematics

Group Theory

34 pages

Scientific paper

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.

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