2-Engel relations between subgroups (Q897754)

Two subgroups \(A\) and \(B\) of a group \(G\) are \textit{\(\mathcal{N}_{2}\)-connected} if, for all \(a \in A\) and \(b \in B\), the group \(\langle a,b \rangle\) is nilpotent of class at most two. This paper is devoted to the study of the groups \(G\) that are generated by two \(\mathcal{N}_{2}\)-connected subgroups \(A\) and \(B\). Under the previous assumption, the authors prove, among other things, that {\parindent=0.8cm \begin{itemize}\item[(i)] \([A,B]\) centralizes \(A'\) and \(B'\), \item[(ii)] \([A',B'] \leq Z(G)\), \item[(iii)] \([A^{2},B]\) is a normal subgroup of \(G\) contained in \(Z([A,B])\), \item[(iv)] \(A'\) is subnormal in \(G\) of defect at most \(3\) and \item[(v)] \(A^{2}\) is subnormal in \(G\) of defect at most \(3\) (here, \(A^{2}=\langle a^{2} \mid a \in A \rangle\)). \end{itemize}}