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 Rpgq such that for every x P G all sufficiently long commutators r...rrg, xs, xs, . . . , xs belong to Rpgq. (Thus, g is a right Engel element precisely when we can choose Rpgq " t1u.) 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 E pgq such that for every x P G all sufficiently long commutators r...rrx, gs, gs, . . . , gs belong to E pgq. (Thus, g is a left Engel element precisely when we can choose E pgq " t1u.) 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.