New paper appeared in “Algebra and Logic”

New paper by Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia) “Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks” has appeared in Algebra and Logic, vol. 62, no. 1 (2023), 80–93.

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 $[\dots [[x,g],g],\dots ,g]$ 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 $[\dots [[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)={ 1}$ .)