Levi classes generated by nilpotent groups

A. I. Budkin

Algebra i logika, Tome 39 (2000) no. 6, pp. 635-647.

Résumé

Let $L(\mathcal M)$ be a class of all groups $G$ for which the normal closure $(x)^G$ of every element $x$ belongs to a class $L(\mathcal M)$ is a Levi class generated by $\mathcal M$. $\mathcal N$ and $\mathcal N_0$ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that $q\mathcal N_0\subset L(q\mathcal N_0)$ and $q\mathcal N\subset L(q\mathcal N)$, and so $L(q\mathcal N_0)\ne qL(\mathcal N_0)$ and $L(q\mathcal N)\ne qL(\mathcal N)$. It is shown that quasivarieties $L(q\mathcal N)$ and $L(q\mathcal N_0)$ are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that $L(\mathcal M)$ is closed under free products if so is $\mathcal M$.