Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks

TL;DR

This paper investigates the structure of finite groups with a soluble automorphism group of coprime order, showing that bounded Engel sinks in the centralizer imply the existence of a subgroup with bounded Fitting height.

Contribution

It establishes bounds on the Fitting height of subgroups in finite groups based on Engel sink properties of elements in the centralizer of a soluble automorphism group.

Findings

Existence of a subgroup with bounded Fitting height depending on automorphism group length and Engel sink bounds.

Bounded Fitting height when elements have bounded rank Engel sinks.

Structural constraints on finite groups with coprime automorphisms and bounded Engel sinks.

Abstract

Suppose that a finite group $G$ admits a soluble group of coprime automorphisms $A$. We prove that if, for some positive integer $m$, every element of the centralizer $C_G(A )$ has a left Engel sink of cardinality at most $m$ (or a right Engel sink of cardinality at most $m$), then $G$ has a subgroup of $(|A|,m)$-bounded index which has Fitting height at most $2\alpha (A)+2$, where $\alpha (A)$ is the composition length of $A$. We also prove that if, for some positive integer $r$, every element of the centralizer $C_G(A )$ has a left Engel sink of rank at most $r$ (or a right Engel sink of rank at most $r$), then $G$ has a subgroup of $(|A|,r)$-bounded index which has Fitting height at most $4^{\alpha (A)}+4\alpha (A)+3$. Here, 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 ${\mathscr E}(g)=\{ 1\}$.) 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 ${\mathscr R}(g)=\{ 1\}$.)