On power margins (Q1120006)

Let \(n\) be an integer and let \(G\) be a group. Then the \(n\)-power margin (in the sense of \textit{P. Hall} [J. Reine Angew. Math. 182, 156-157 (1940; Zbl 0023.29902)]) of \(G\) is the characteristic subgroup \(M_ n(G)=\{a\in G\mid (ag)^ n=g^ n\) for all \(g\in G\}\). Moreover, let \(R_ n[m]=\{a\in G\mid a^ m=1\) and \([a,_ nx]=1\) for all \(x\in G\}\) be the set of all right \(n\)-Engel elements in \(G\) of order dividing \(m\). It is shown that \(M_ 3(G)=R_ 2[3]\) and \(M_ 3(G)\leq Z_ 3(G)\) and if \(G\) is metabelian and \(p\) is a prime, then \(M_ p(G)=R_{p-1}[p]\) and \(M_ p(G)\leq Z_ p(G)\); the bounds being best possible. Also, for finite groups, the \(p^ m\)-power margin is contained in the hypercentre.