Finite and profinite groups with small engel sinks of p-elements

Abstract

A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators , where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N, such that G/N is a -group and the index is bounded in terms of m only. Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p, while G/N is a pro- group.