[Paper Review] Compact groups all elements of which are almost right Engel

E. I. Khukhro, Pavel Shumyatsky|arXiv (Cornell University)|Jul 14, 2018

Finite Group Theory Research14 references4 citations

TL;DR

This paper proves that in a compact Hausdorff group where every element is almost right Engel—meaning repeated right commutators eventually enter a finite set—there exists a finite normal subgroup such that the quotient is locally nilpotent. The key contribution is a structural classification of such compact groups, extending earlier results on almost left Engel elements and establishing quantitative bounds when the Engel sinks are uniformly bounded in size.

ABSTRACT

We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[g,x],x],\dots ,x]\in {\mathscr R}(g)$ if $x$ is repeated at least $n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$. We prove that if all elements of a compact (Hausdorff) group $G$ are almost right Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a uniform bound $|{\mathscr R}(g)|\leq m$ for the orders of the corresponding sets, then the subgroup $N$ can be chosen of order bounded in terms of $m$. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.

Motivation & Objective

• To classify compact groups in which every element is almost right Engel, generalizing prior results on almost left Engel elements.
• To establish the existence of a finite normal subgroup with locally nilpotent quotient under the almost right Engel condition.
• To provide quantitative bounds on the order of the finite normal subgroup when the size of the right Engel sink is uniformly bounded across all group elements.

Proposed method

• Use of the Wilson–Zelmanov theorem on profinite Engel groups being locally nilpotent as a foundational tool.
• Reduction of the problem to finite groups via inverse limit structures and profinite group theory.
• Application of structure theorems for compact groups to lift results from profinite to general compact groups.
• Leveraging minimal Engel sinks and their conjugacy properties to control group actions on finite quotients.
• Use of Maschke’s theorem and properties of $p$-groups to analyze automorphism actions on quotients.
• Employment of inverse limit arguments to extend finite group results to the compact group setting.

Experimental results

Research questions

• RQ1 Does every compact group in which all elements are almost right Engel necessarily have a finite normal subgroup with locally nilpotent quotient?
• RQ2 Can the order of the finite normal subgroup be bounded in terms of the maximum size of the right Engel sinks when this size is uniformly bounded?
• RQ3 Is there a structural connection between almost right Engel and almost left Engel elements in compact groups, and if so, what does it imply?
• RQ4 How do the properties of Engel sinks in finite groups extend to profinite and compact groups?
• RQ5 What role do $p$-groups and Frattini quotients play in controlling the action of elements on finite quotients in this context?

Key findings

• Every compact group in which all elements are almost right Engel has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.
• If the size of the right Engel sink ${\mathscr{R}}(g)$ is uniformly bounded by $m$ for all $g \in G$, then the finite normal subgroup $N$ can be chosen of order bounded in terms of $m$.
• The result implies that such compact groups are finite-by-(locally nilpotent), extending the finite-by-nilpotent structure known from almost left Engel elements.
• All elements in such a group also have finite left Engel sinks, so the group satisfies the conditions of the authors' earlier result on almost left Engel elements.
• The proof uses inverse limit techniques and structure theorems for compact groups, reducing the problem to finite and profinite groups.
• The nilpotent residual $\gamma_\infty(H)$ of any finitely generated subgroup $H$ is finite of $m$-bounded order when $|{\mathscr{R}}(g)| \leq m$ uniformly.