Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

2021/01/09 by Khukhro, E. I., Shumyatsky, P. · 1 citation
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2101.03404

Abstract

A right Engel sink of an element g of a group G is a set \mathscr R(g) such that for every x∈ G all sufficiently long commutators [...[[g,x],x],… ,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 profinite group G admits a coprime automorphism φ of prime order such that every fixed point of φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set \mathscr E(g) such that for every x∈ G all sufficiently long commutators [...[[x,g],g],… ,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 profinite group G admits a coprime automorphism φ of prime order such that every fixed point of φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

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