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

Khukhro, Evgeny and Shumyatsky, Pavel (2021) On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks. Bulletin of the Brazilian Mathematical Society . ISSN 1678-7544

Full content URL: https://doi.org/10.1007/s00574-021-00249-6

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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 $m$.