  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.

Group Theory