Dense subclasses in some varieties of two-step nilpotent groups

P. M. Beletskii

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 369-385

Résumé

Group varieties $\mathfrak W=\mathfrak W(p,k)$ defined by the identities $$ x^{p^{2k}}=1,\qquad[x,y]^{p^k}=1,\qquad[x,y,z]=1 $$ ($p$ is a prime) are examined. For $\mathfrak W'\subseteq\mathfrak W$ the set of all pairwise nonisomorphic $n$-generated groups in $\mathfrak W'$ is denoted by $\mathfrak W_n'$, and the subclass $\mathfrak W'$ is called dense in $\mathfrak W$ if $|\mathfrak W_n'|/|\mathfrak W_n|\to1$ as $n\to\infty$. A general method of investigating the numerical sequence $\{|\mathfrak W_n'|/|\mathfrak W_n\vert\}$ is presented. In particular, it is proved that the subclass of groups with an abelian automorphism group is dense in the variety $\mathfrak W(p,k)$. Bibliography: 14 titles.