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Almost Engel finite and profinite groups

E. I. Khukhro

School of Mathematics and Physics, University of Lincoln, Lincoln, Brayford Pool, LN6 7TS, UK

Sobolev Institute of Mathematics, Novosibirsk 630090, Russia

E-mail Address: khukhro@yahoo.co.uk

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and

P. Shumyatsky

Department of Mathematics, University of Brasilia, Brasilia, DF 70910-900, Brazil

E-mail Address: pavel@unb.br

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https://doi.org/10.1142/S0218196716500405Cited by:7 (Source: Crossref)

Abstract

Let $g$ $g$ be an element of a group $G$ $G$ . For a positive integer $n$ $n$ , let $E_{n} (g)$ $E_{n} (g)$ be the subgroup generated by all commutators $[\dots [[x, g], g], \dots, g]$ $[\dots [[x, g], g], \dots, g]$ over $x \in G$ $x \in G$ , where $g$ $g$ is repeated $n$ $n$ times. We prove that if $G$ $G$ is a profinite group such that for every $g \in G$ $g \in G$ there is $n = n (g)$ $n = n (g)$ such that $E_{n} (g)$ $E_{n} (g)$ is finite, then $G$ $G$ has a finite normal subgroup $N$ $N$ such that $G / N$ $G / N$ is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$ $G$ , we prove that if, for some $n$ $n$ , $| E_{n} (g) | \leq m$ $| E_{n} (g) | \leq m$ for all $g \in G$ $g \in G$ , then the order of the nilpotent residual $γ_{\infty} (G)$ $γ_{\infty} (G)$ is bounded in terms of $m$ $m$ .

Communicated by A. Olshansky

Keywords:

AMSC: 20D25, 20E18, 20F45

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