Engel sinks of fixed points in finite groups

Abstract

For an element $g$ of a group $G$, an Engel sink is a subset $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[x,g,g,\dots ,g]$ belong to $\mathcal{E}(g)$. 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 $\mathcal{E}(g)$ 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 $\mathcal{E}(g)$ generates a subgroup of rank at most $m$, then the rank of ${\gamma }_{\infty }(G)$ is bounded in terms of $m$ and $q$ only.