Profinite groups and centralizers of coprime automorphisms whose elements are Engel

TL;DR

This paper investigates the structure of finite and profinite groups with automorphisms of coprime order, showing that certain Engel conditions on centralizers imply bounded Engel properties on derived and lower central series.

Contribution

It establishes new bounds on Engel properties of groups based on conditions on centralizers under coprime automorphisms, extending to profinite groups.

Findings

Bounded Engel properties for derived subgroups under centralizer conditions

Results apply to both finite and profinite groups

Provides explicit bounds depending on parameters q, r, n

Abstract

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $\gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $\gamma_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$ we prove that if all elements in $\gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#$, then $\gamma_{r-2}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.