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Dense subclasses in some varieties of two-step nilpotent groups
P. M. Beletskii
Abstract:
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.
Received: 10.11.1982
Citation:
P. M. Beletskii, “Dense subclasses in some varieties of two-step nilpotent groups”, Mat. Sb. (N.S.), 122(164):3(11) (1983), 388–404; Math. USSR-Sb., 50:2 (1985), 369–385
Linking options:
https://www.mathnet.ru/eng/sm2300https://doi.org/10.1070/SM1985v050n02ABEH002834 https://www.mathnet.ru/eng/sm/v164/i3/p388
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Abstract page: | 216 | Russian version PDF: | 57 | English version PDF: | 6 | References: | 34 |
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