Centralizers of coprime automorphisms of finite groups

TL;DR

This paper investigates how the nilpotency properties of centralizers of automorphisms influence the nilpotency of the entire finite group, extending known results to higher cases with bounded nilpotency class.

Contribution

It generalizes previous results by establishing bounds on the nilpotency class of the group based on properties of centralizers for larger values of k.

Findings

If certain centralizers are nilpotent of bounded class, then the whole group is nilpotent with bounded class.

The results extend to higher derived subgroups under specified conditions.

Provides bounds on nilpotency class depending on parameters p, k, c.

Abstract

Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then $\gamma_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$.