Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks

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 C G (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 C G (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 4 α(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}.