Almost Engel finite and profinite groups (Q2821824)

A group \(G\) is called an Engel group if for every \(x, g \in G\) there is an integer \(n\), depending on \(x\) and \(g\), such that the repeated commutator \([x, \underbrace{g, \dots, g}_{n}]\) vanishes. A group is said to be locally nilpotent if every finitely generated subgroup is nilpotent. Clearly, a locally nilpotent group is an Engel group. \textit{J. S. Wilson} and \textit{E. I. Zelmanov} have proved that the converse holds for profinite groups [J. Pure Appl. Algebra 81, No. 1, 103--109 (1992; Zbl 0851.17007)].NEWLINENEWLINEThe main result of the paper (Theorem 1.1) deals with a profinite group \(G\) such that for every \(g \in G\), there is an \(n\), depending on \(g\), such that the subgroup \(E_{n}(G) = \langle [x, \underbrace{g, \dots, g}_{n}] : x \in G \rangle\) is finite. Such a group is shown to be finite-by-(locally nilpotent).NEWLINENEWLINEThe result admits a quantitative version for finite groups. Let \(E = \cap_{n=1}^{\infty} E_{n}(G)\). Theorem~1.2 states that if \(G\) is a finite group such that there is a bound \(m\) on the size of the \(E(g)\), for \(g \in G\), then the order of the nilpotent residual of \(G\) is bounded in terms of \(n\). A natural consequence for profinite groups is derived.