Engel sinks of fixed points in finite groups

TL;DR

This paper investigates the structure of finite groups with certain constraints on Engel sinks of elements fixed by an elementary abelian group, establishing bounds on the order and rank of the group's nilpotent residual.

Contribution

It provides new bounds on the order and rank of the nilpotent residual in finite groups based on properties of Engel sinks under coprime group actions.

Findings

Bounded the order of the nilpotent residual by the size of Engel sinks.

Bounded the rank of the nilpotent residual by the size of Engel sinks and the order of the acting group.

Established conditions under which the structure of the nilpotent residual is controlled by Engel sink properties.

Abstract

For an element $g$ of a group $G$, an Engel sink is a subset $\mathscr{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $g\in C_{G}(a)$ the cardinality of the smallest Engel sink $\mathscr{E}(g)$ is at most $m$, then the order of $\gamma_\infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $a\in A\setminus \{1\}$ and every element $g\in C_{G}(a)$, the smallest Engel sink $\mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $\gamma_\infty(G)$ is bounded in terms of $m$ and $q$ only.