Nilpotent residual of fixed points

TL;DR

This paper proves that the nilpotent residual of a finite group acted upon by a prime order automorphism group is bounded by parameters related to fixed point centralizers, extending understanding of group structure under automorphisms.

Contribution

It establishes bounds on the nilpotent residual of a finite group based on the properties of fixed point centralizers under automorphisms of prime order.

Findings

Bound on the order of the nilpotent residual in terms of fixed point centralizer size.

Bound on the rank of the nilpotent residual in terms of fixed point centralizer rank.

Results depend only on parameters related to the automorphism group and fixed points.

Abstract

Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q^3$. We show that if $\gamma_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in A^{\#}$, then the order of $\gamma_{\infty} (G)$ is bounded solely in terms of $m$ and $q$. If $\gamma_{\infty} (C_{G}(a))$ has rank at most $r$ for any $a \in A^{\#}$, then the rank of $\gamma_{\infty} (G)$ is bounded solely in terms of $r$ and $q$.