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Mat. Sb., 2015, Volume 206, Number 3, Pages 143–152 (Mi msb8353)  

Distributions of elements on nilpotent varieties of groups

E. I. Timoshenko

Novosibirsk State Technical University

Abstract: Let $\mathfrak M$ be some variety of groups, and $F_n(\mathfrak M)$ a free group in $\mathfrak M$ with a basis $\{x_1,…,x_n\}$. Two elements $u(x_1,…,x_n)$ and $v(x_1,…,x_n)$ of this group induce the same distributions on $\mathfrak M$ if for any finite group $G\in\mathfrak M$ and any element $g\in G$ the equations $u(x_1,…,x_n)=g$ and $v(x_1,…,x_n)=g$ have the same number of solutions in $G^n$.
It is proved that two elements of the derived subgroup of a free group of the variety of nilpotent groups of class at most 2 induce the same distributions on this variety if and only if these elements can be transformed into each other by automorphisms, but this is not true for elements that do not belong to the derived subgroup.
Bibliography: 5 titles.

Keywords: variety of groups, nilpotent groups, equations in groups, distributions of elements.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-01485
Ministry of Education and Science of the Russian Federation 14.В37.21.0359
This research was supported by the Russian Foundation for Basic Research (grant no. 15-01-01485) and by a Programme of the Ministry for Education and Science of the Russian Federation (grant no. 14.В37.21.0359).


DOI: https://doi.org/10.4213/sm8353

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English version:
Sbornik: Mathematics, 2015, 206:3, 470–479

Bibliographic databases:

UDC: 512.544.33+512.543.5
MSC: Primary 20F18; Secondary 20P05
Received: 04.03.2014 and 26.09.2014

Citation: E. I. Timoshenko, “Distributions of elements on nilpotent varieties of groups”, Mat. Sb., 206:3 (2015), 143–152; Sb. Math., 206:3 (2015), 470–479

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