# Algebra in Lincoln

School of Mathematics & Physics, University of Lincoln

# New paper

E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups, submitted, 2015; 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 $m$.

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This entry was posted on January 4, 2016 by in New publications, News and announcements.