Finite Groups with a Soluble Group of Coprime Automorphisms Whose Fixed Points Have Bounded Engel Sinks

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 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 (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}.