On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks

Abstract

A left Engel sink of an element $g$ of a group $G$ is a set $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to $\mathcal{E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose $\mathcal{E}(g)=\{1\}$.) We prove that if a finite group $G$ admits an automorphism $\phi$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\phi )$ has a left Engel sink of cardinality at most $m$, then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of $m$. A right Engel sink of an element $g$ of a group $G$ is a set $\mathcal{R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to $\mathcal{R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose $\mathcal{R}(g)=\{1\}$.) We prove that if a finite group $G$ admits an automorphism $\phi$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\phi )$ has a right Engel sink of cardinality at most $m$, then the index of the Fitting subgroup $F_1(G)$ is bounded in terms of $m$.