Two generator \(p\)-groups of nilpotency class 2 and their conjugacy classes. (Q2915426)

For a prime \(p\) the authors classify the 2-generated groups of order \(p^n\) and of nilpotency class 2 by assigning a 5-tuple \((\alpha,\beta,\gamma;\rho,\sigma)\) to each of them such that (i) \(\alpha\geq\beta\geq\gamma\geq 1\), (ii) \(\alpha+\beta+\gamma=n\), (iii) \(0\leq\rho,\sigma\leq\gamma\), where \((\alpha,\beta,\gamma;\rho,\sigma)\) corresponds to the group NEWLINE\[NEWLINEG=\langle a,b\mid [a,b]^{p^\gamma}=[[a,b],a]=[[a,b],b]=1,\;a^{p^\alpha}=[a, b]^{p^\rho},\;b^{p^\beta}=[a,b]^{p^\sigma}\rangle.NEWLINE\]NEWLINE Moreover, they determine which 5-tuples are assigned to isomorphic groups by dividing them into 7 families (Theorem~1.1). As an application, they prove in Theorem~1.2 that a 2-generated group of order \(p^n\) and with derived subgroup of order \(p^\gamma\) has exactly \(p^{n-\gamma}(1+p^{-1}-p^{-(\gamma+1)})\) conjugacy classes, which, in turn, gives a formula on the abundance of such a group.NEWLINENEWLINE \textit{M. R. Bacon} and \textit{L.-C. Kappe} [Arch. Math. 61, No. 6, 508-516 (1993; Zbl 0823.20021)] considered the tensor squares of 2-generated \(p\)-groups of nilpotency class 2 for odd primes \(p\), for which they classified all such groups. Their results were extended by \textit{L.-C. Kappe, M. P. Visscher} and \textit{N. H. Sarmin} [Glasg. Math. J. 41, No. 3, 417-430 (1999; Zbl 0970.20012)] for the case \(p=2\). The authors of the paper under review devote Section~6 to show the similarities and differences between their families of groups and those in these two papers, claiming that the earlier characterizations miss one family of groups, and hence are incomplete. Furthermore, the authors compare their families of groups to those of \textit{R. J. Miech} [in J. Aust. Math. Soc., Ser. A 20, 178-198 (1975; Zbl 0319.20024)] classifying 2-generated \(p\)-groups with cyclic commutator subgroup.