On a problem of P. Hall for Engel words. II. (Q2922936)

The author considers Engel words \([[x,y,\ldots,y],y]\) of length three and four, the subgroups \(E_n(G)\) generated by these Engel words, the sets of right and left Engel elements \(R_n(G)\), \(L_n(G)\), its intersection \(T_n(G)\), and the marginal subgroups \(E_n^*(G)\).NEWLINENEWLINE The derived length of \(G/E_n(G)\) and \(E_n^*(G)\) is not bounded in general, but if it is bounded estimates for the length of the central series are given (Theorem 1.1). Restriction to groups of odd order allows to locate \(T_3(G)\) in \(\mathrm{Fit}(G)\) (Theorem 1.2). Descriptions for partial marginal subgroups (where only one variable is allowed to be variated) are also given.NEWLINENEWLINE For part I of this series see \textit{A. Abdollahi} and the author [Arch. Math. 97, No. 5, 407-412 (2011; Zbl 1236.20044)].