A construction of metabelian groups. (Q1779988)

Under metabelian groups the author understands the groups of class \(2\). It was shown by \textit{G. Birkhoff} [Proc. Lond. Math. Soc., II. Ser. 38, 385-401 (1934; Zbl 0010.34304)], that the number of isomorphism classes of groups of order \(p^{22}\) and nilpotence class \(2\) tends to infinity with \(p\). In this note the author uses recent results on the classification of possible embeddings of a subgroup in a finite Abelian \(p\)-group to construct families of indecomposable groups of class \(2\), indexed by several parameters, which have upper bounds on the exponents of the center and the derived subgroup.