On profinite groups in which commutators are Engel (Q2726646)

Let \(G\) be a group and \(x,y\) be elements of \(G\). Denote \([x_1,x_2,\dots,x_k]=[[x_1,x_2,\dots,x_{k-1}],x_k]\) and \([x,{_0y}]=x\) and \([x,{_ny}]=[[x,{_{n-1}y}],y]\) for positive integers \(k,n\). An element \(y\) of \(G\) is called Engel if for any \(x\in G\) there exists a natural number \(n\) such that \([x,{_ny}]=1\). A group \(G\) is Engel if every element of \(G\) is Engel. \textit{J. S. Wilson} and \textit{E. I. Zelmanov} [in J. Pure Appl. Algebra 81, No. 1, 103-109 (1992; Zbl 0851.17007)] proved that any profinite Engel group is locally nilpotent. The author proves the following two results.NEWLINENEWLINENEWLINETheorem 1.1. Let \(k\) be a positive integer and \(G\) a finitely generated profinite group such that \([x_1,x_2,\dots,x_k]\) is Engel for all \(x_1,\dots,x_k\in G\). Then the \(k\)-th term of lower central series \(\gamma_k(G)\) is locally nilpotent.NEWLINENEWLINENEWLINETheorem 1.2. Let \(k\) be a positive integer and \({\mathcal N}^h\) a class of finite solvable groups of Fitting heigth \(h\). Let \(G\) be a finitely generated pro-\({\mathcal N}^h\) group such that \([x_1,x_2,\dots,x_k]\) is of finite order for all \(x_1,\dots,x_k\in G\). Then the \(k\)-th term of lower central series \(\gamma_k(G)\) is locally finite.NEWLINENEWLINENEWLINEThe methods of proofs are based on Lie ring methods.