On a problem of P. Hall for Engel words. (Q652229)

Let \([x,y]=[x,{_1y}]\) and \([[x,{_ky}],y]=[x,{_{k+1}y}]\). In this article the word \(e_n(x,y)=[x,{_ny}]\) and the corresponding set \(\{E_n\}(G)\) of words, the word subgroups \(E_n(G)\) and the marginal subgroups \(E^*(G)\) are considered. Results: If \(\{E_2\} (G)\) is finite, then \(E_2(G)\) is finite; further \(G/E_2^*(G)\) is finite if \(G\) is finitely generated (Theorem 2.3). -- If \(k\in\{3,4\}\) and \(\{E_k\}(G)\) is finite, then \(E_k(G)\) is finite (Theorem 2.4). Inequalities on the order and the number of generators of \(G/E_2^*(G)\) are given.