FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 3, PAGES 871-883

On full residuance of some groups

P. V. Ushakov

Abstract

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In this paper we define the rank of a finitely generated torsion free nilpotent group. The main result is the following

Theorem. Let $G$ be a finitely generated nilpotent group. Let $\$\; \backslash mathfrak\; U\; \$$ be an arbitrary variety of groups. Assume that $G$ is torsion free, $\$\; \backslash mathop\{\backslash mathrm\{rk\}\}\; G=k\; \$$, $\$\; \backslash mathfrak\; N\; :=\; \backslash mathop\{\backslash mathrm\{var\}\}\; G\; \$$, $G$@ F_k/R, $\$\; R\; \backslash triangleleft\; F\_k\; \$$. Then for all $s\; >\; k$, the groups $\$\; F\_s(\backslash mathfrak\; \{UN\})\; \$$ are fully residual $F$_k/U(R)-groups.

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Last modified: November 11, 1999