On the power-commutative kernel of locally nilpotent groups. (Q2368389)

A group is called power-commutative, or a PC-group, if \([x^m,y^n]=1\) implies \([x,y]=1\) for all \(x,y\in G\) such that \(x^m\neq 1\), \(y^n\neq 1\). The authors of the paper under review, in analogy to what was done by \textit{B. Fine, A. M. Gaglione, G. Rosenberger}, and \textit{D. Spellman} [in Algebra Colloq. 4, No. 2, 141-152 (1997; Zbl 0901.20011)] to define the commutative-transitive kernel of a group, introduce an ascending series \[ \{1\}=P_0(G)\leq P_1(G)\leq\cdots\leq P_t(G)\leq\cdots \] of characteristic subgroups of \(G\) contained in the derived subgroup \(G'\). The authors define \(P_1(G)\) as the subgroup of \(G'\) generated by those commutators \([x,y]\) such that there exist positive integers \(n,m\) with \(x^n\neq 1\), \(y^m\neq 1\), and \([x^n,y^m]=1\). If \(t>1\), then \(P_t(G)\) is defined by \(P_t(G)/P_{t-1}(G)=P_1(G/P_{t-1}(G))\). Now, the PC-kernel of \(G\) is the subgroup \(P(G)\) of \(G'\) defined by \(P(G)=\bigcup_{t\in\mathbb{N}}P_t(G)\). For any group \(G\), the PC-kernel \(P(G)\) is characteristic in \(G\), \(G/P(G)\) is a PC-group, and \(G\) is a PC-group if and only if \(P(G)=\{1\}\). The authors consider the following question: Let \(\mathcal X\) be a class of groups. Does there exist a nonnegative integer \(n\) such that \(P_n(G)=P(G)\) for all \(G\in{\mathcal X}\)? In the paper under review, the authors give affirmative answers to the previous question when \(\mathcal X\) is the class of locally nilpotent groups, or the class of finite groups having nontrivial center. In both cases, they prove that \(P(G)=P_1(G)\) for all \(G\in{\mathcal X}\).