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

Abstract

A right Engel sink of an element g of a group G is a set {{mathscr {R}}}(g) such that for every xin 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 profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi 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 xin 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 profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.