Repeated and final commutators in group actions

Abstract:

Let $G$ be a finite group and suppose that $A$ acts via automorphisms on $G$. The repeated commutators are the subgroups $[G,A,A,\ldots ,A]$, where there is some positive number of commutations by $A$, and the final commutator is the smallest of these repeated commutators. We show that if $[G,A]$ is nilpotent, then the final commutator is normal in $G$. Also, in general, if $K$ is an arbitrary repeated commutator and $P$ is the permutation group induced by the action of $A$ on the left cosets of $K$ in $G$, we relate the structure of $P$ to the structure of $[G,A]$.