Abstract

Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer CG(A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of (|A|,m)-bounded index which has Fitting height at most 2α(A)+2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer CG(A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of (|A|,r)-bounded index which has Fitting height at most 4α(A)+4α(A)+3. Here, a left Engel sink of an element g of a group G is a set E(g) such that for every x∈G all sufficiently long commutators [...[[x,g],g],…,g] belong to E(g). (Thus, g is a left Engel element precisely when we can choose E(g)={1}.) A right Engel sink of an element g of a group G is a set R(g) such that for every x∈G all sufficiently long commutators [...[[g,x],x],…,x] belong to R(g). (Thus, g is a right Engel element precisely when we can choose R(g)={1}.)