Groups with an automorphism of prime order that is almost regular in the sense of rank

Khukhro, E. I. (2008) Groups with an automorphism of prime order that is almost regular in the sense of rank. J. London Math. Soc., 77. pp. 130-148.

Let $\varphi$ be an automorphism of prime order $p$ of a finite group $G$, and let $r$ be the (Pr\"ufer) rank of the fixed-point subgroup $C_G(\varphi )$. It is proved that if $G$ is nilpotent, then there exists a characteristic subgroup $C$ of nilpotency class bounded in terms of $p$ such that the rank of $G/C$ is bounded in terms of $p$ and $r$. For infinite (locally) nilpotent groups a similar result holds if the group is torsion-free (due to Makarenko), or periodic, or finitely generated; but examples show that these additional conditions cannot be dropped, even for nilpotent groups. As a corollary when $G$ is an arbitrary finite group, the combination with the recent theorems of the author and Mazurov gives characteristic subgroups $R\leqslant N\leqslant G$ such that $N/R$ is nilpotent of class bounded in terms of $p$, while the ranks of $R$ and $G/N$ are bounded in terms of $p$ and $r$ (under the additional unavoidable assumption that $p\nmid |G|$ if $G$ is insoluble); in general it is impossible to get rid of the subgroup~$R$. The inverse limit argument yields corresponding consequences for locally finite groups.