Levi classes of quasivarieties of nilpotent groups of exponent $p^s$

V. V. Lodeishchikova ; S. A. Shakhova

Algebra i logika, Tome 61 (2022) no. 1, pp. 77-92

Résumé

The Levi class $L(\mathcal{M})$ generated by the class $\mathcal{M}$ of groups is the class of all groups in which the normal closure of every element belongs to $\mathcal{M}$. It is proved that there exists a set of quasivarieties $\mathcal{M}$ of cardinality continuum such that $L(\mathcal{M})=L(qH_{p^{s}})$, where $qH_{p^{s}}$ is the quasivariety generated by the group $H_{p^{s}}$, a free group of rank $2$ in the variety $\mathcal{R}^{p^{s}}$ of $\leq 2$-step nilpotent groups of exponent $p^{s}$ with commutator subgroup of exponent $p$, $p$ is a prime number, $p\neq 2$, $s$ is a natural number, $s\geq 2$, and $s>2$ for $p=3$.

Détail

Keywords: quasivariety, Levi class, nilpotent group.