Paper accepted for publication

The paper by E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups has been accepted for publication in the International Journal of Algebra and Computation, ISSN 0218-1967, see also arXiv:1512.06097.

Abstract: Let $g$ be an element of a group $G$ . For a positive integer $n$ , let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots, g]$ over $x\in G$ , where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ 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 $G$ , we prove that if, for some $n$ , $|E_n(g)|\leq m$ for all $g\in G$ , then the order of the nilpotent residual $\gamma _{\infty} (G)$ is bounded in terms of $m$ .