On Finite Groups with an Automorphism of Prime Order Whose Fixed Points Have Bounded Engel Sinks

A left 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 a left Engel element precisely when we can choose E(g) = {1}.) We prove that if a finite group G admits an automorphism phi of prime order coprime to vertical bar G vertical bar such that for some positive integer m every element of the centralizer C-G (phi) has a left Engel sink of cardinality at most m, then the index of the second Fitting subgroup F-2 (G) is bounded in terms of m. A right Engel sink of an element g of a group G is a set R(g) such that for every x is an element of 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}.) We prove that if a finite group G admits an automorphism phi of prime order coprime to vertical bar G vertical bar such that for some positive integer m every element of the centralizer C-G(phi) has a right Engel sink of cardinality at most m, then the index of the Fitting subgroup F-1(G) is bounded in terms of in.