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 16787544
Full content URL: https://doi.org/10.1007/s00574021002496
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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 $m$.
Keywords:  Finite groups, Engel condition, Fitting subgroup, automorphism 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  44077 
Deposited On:  22 Feb 2021 12:41 
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