On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks

Abstract

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}.) We prove that if a finite group G admits an automorphism ϕ of prime order coprime to |G| such that for some positive integer m every element of the centralizer C G (ϕ) 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 ∈ 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 ϕ of prime order coprime to |G| such that for some positive integer m every element of the centralizer C G (ϕ) 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 m.

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

“…The proof also makes use of the quantitative version for finite groups that we proved earlier in [12]. In that paper [12] we also proved that if a finite group G has a coprime automorphism ϕ of prime order such that all fixed points of ϕ have left Engel sinks of cardinality at most m, then G has a metanilpotent subgroup of index bounded in terms of m (examples show that here "metanilpotent" cannot be replaced by "nilpotent"). We prove the following profinite analogue of this result.…”

“…Similarly to Example 5.2, using finite direct products H " ś n i"1 V i ¸pxa i y xb i yq instead of the Cartesian product, we obtain examples of finite groups G with a coprime automorphism ϕ of order 2 such that all elements of C G pϕq have trivial left Engel sinks. These examples show that the conclusion of [12,Theorem 1.3] giving a bound for the index of F 2 pGq cannot be improved to a bound for the index of F pGq.…”

“…One of the important tools in the proof of Theorem 1.2 is a strengthened version of Neumann's theorem about BF C-groups from the recent paper of Acciarri and Shumyatsky [1]. The proof also makes use of the quantitative version for finite groups that we proved earlier in [12]. In that paper [12] we also proved that if a finite group G has a coprime automorphism ϕ of prime order such that all fixed points of ϕ have left Engel sinks of cardinality at most m, then G has a metanilpotent subgroup of index bounded in terms of m (examples show that here "metanilpotent" cannot be replaced by "nilpotent").…”