ON PROFINITE GROUPS WITH AUTOMORPHISMS WHOSE FIXED POINTS HAVE COUNTABLE ENGEL SINKS

An Engel sink of an element g of a group G is a set E(g) such that for every x is an element of G all sufficiently long commutators [... [[x, g], g], ... , g] belong to E(g). (Thus, g is an Engel element precisely when we can choose E (g) = {1}.) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q(2) for a prime q such that for each a is an element of A \ {1} every element of the centralizer C-G(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.