Fitting subgroup and nilpotent residual of fixed points

TL;DR

This paper investigates the structure of finite groups under automorphisms, establishing bounds on the Fitting subgroup and nilpotent residual based on fixed point subgroup properties.

Contribution

It proves bounds on the Fitting subgroup and nilpotent residual of a finite group using automorphism fixed point conditions, extending previous structural results.

Findings

Bound on the order of the nilpotent residual based on fixed points

Bound on the index of the second Fitting subgroup under automorphism conditions

Results apply to groups with elementary abelian automorphism groups

Abstract

Let $q$ be a prime and $A$ an elementary abelian group of order at least $q^3$ acting by automorphisms on a finite $q'$-group $G$. It is proved that if $|\gamma_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\gamma_{\infty}(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_G(a)$ for any $a \in A^{\#}$, then the index of $F_2(G)$ is $m$-bounded.