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

TL;DR

The paper proves that finite groups with a prime order automorphism and bounded Engel sinks in the centralizer have bounded indices of their Fitting subgroups, linking automorphism properties to group structure constraints.

Contribution

It establishes bounds on the indices of Fitting subgroups in finite groups based on the boundedness of Engel sinks for elements fixed by a prime order automorphism, a novel structural insight.

Findings

Bounded index of the second Fitting subgroup when Engel sinks are bounded.

Bounded index of the Fitting subgroup when Engel sinks are bounded.

Automorphism of prime order influences the group's Fitting subgroup structure.

Abstract

A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $\varphi $ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ 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 ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $\varphi$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ 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 $m$.