On profinite groups with automorphisms whose fixed points have countable Engel sinks

Abstract

An Engel sink of an element g of a group G is a set $${\cal E}(g)$$ such that for every x ∈ G all sufficiently long commutators [⋯[[x, g], g],…, g] belong to $${\cal E}(g)$$ . (Thus, g is an Engel element precisely when we can choose $${\cal E}(g) = \{ 1\} $$ .) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q2 for a prime q such that for each a ∈ A {1} every element of the centralizer CG(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.