2012.93: Groups with an automorphism of prime order that is almost regular in the sense of rank

2012.93: E. I. Khukhro (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.

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Abstract

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.

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