On groups with automorphisms whose fixed points are Engel

Abstract

Abstract. We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems.Let q be a prime and A an elementary abelian q-group of order at least q 2 acting coprimely on a profinite group G. Assume that all elements in C G (a) are Engel in G for each a ∈ A # . Then G is locally nilpotent (Theorem B2).Let q be a prime, n a positive integer and A an elementary abelian group of order q 3 acting coprimely on a finite group G. Assume that for each a ∈ A # every element of C G (a) is n-Engel in C G (a). Then the group G is k-Engel for some {n, q}-bounded number k (Theorem A3).

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

“…Some of the best-known examples include Thompson's theorem [23] on the nilpotency of a finite group with a fixedpoint-free automorphism of prime order, as well as numerous other papers on finite groups admitting automorphisms with various restrictions on their fixed points. This type of results in relation to automorphisms whose fixed points have restrictions on their Engel sinks were recently obtained in [1,2,3,4,5,20].…”

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

“…A profinite (non-quantitative) version of the above theorem was established in the recent work [4]. Theorem 1.4.…”

“…An example of a finite non-nilpotent group G admitting an action of a noncyclic group A of order four such that C G (a) is abelian for each a ∈ A # can be found for instance in [3]. On the other hand, another result, that was established in [4], is the following. Theorem 1.5.…”

Engel-like conditions in fixed points of automorphisms of profinite groups

“…. ∈ L 1 are common eigenvectors for A generating L, then any commutator in those generators belongs to some L ij and so, by (2), is ad-nilpotent.…”

“…We say that a group A acts on a profinite group G coprimely if A has finite order while G is an inverse limit of finite groups whose orders are relatively prime to the order of A. In the literature there are many well-known results showing that if A is a finite group acting on a finite group G in such a manner that (|A|, |G|) = 1, then the structure of the centralizer C G (A) (the fixed-point subgroup) of A has a strong influence over the structure of G (see for instance [2,10,21,22]). A similar phenomenon holds in the realm of profinite groups: we see that imposing restrictions on centralizers of coprime automorphisms result in very specific structures for the group G. Given an automorphism a of a profinite group G, we denote by C G (a) the centralizer of a in G, that is, the subgroup formed by the elements fixed under a.…”