Diagonal embeddings of nilpotent groups
Let G be a group, \(G^{\phi}\) an isomorphic copy of G, where \(\phi\) is a fixed isomorphism and \(g^{\phi}\) is the image of \(g\in G\) under \(\phi\). A double group D(G) is a group with the presentation \(D(G)=<G,G^{\phi}\); \([g,g^{\phi}]=1\), for all \(g\in G>\). Double groups have been studied before by two of the authors [\textit{N. Rocco}, J. Algebra 76, 471-488 (1982; Zbl 0489.20018)] and \textit{S. Sidki} [ibid. 63, 186-225 (1980; Zbl 0442.20014)]. In this paper the authors give a description of the lower central series of D(G) in terms of the lower central series of G. After ingeniously developing a commutator calculus for D(G) they prove the following results: (1) if G is locally nilpotent, then D(G) is locally nilpotent (2) if G is a 2-generator nilpotent group of class at most c, then D(G) is nilpotent of class at most \(c+1\). (3) If G is an m-generator nilpotent group of class at most c with \(m\geq 2\), \(c\geq 1\), then for \(m\leq c+2\), \(\gamma_{c+3}(D(G))=1\) while for \(m\leq c+3\), \(\gamma_{c+3}(D(G))\) is an elementary abelian 2-group of rank at most \(\sum^{m}_{k=c+3}\left( \begin{matrix} m\\ k\end{matrix} \right)\). Here \(\gamma_ i(H)\) denotes, as usual, the i-th term of the lower central series of the group H.
scientific article; zbMATH DE number 3950838