# New paper accepted by Bulletin of the Brazilian Mathematical Society

New paper by **Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia)** “On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks” has been accepted for publication in the *Bulletin of the Brazilian Mathematical Society*. The work was supported by Mathematical Center in Akademgorodok, FAPDF and CNPq-Brazil, and stems from the collaboration with University of Brasilia.

*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 $m$.

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