On the complexity of the lattice of quasivarieties of nilpotent groups

A. I. Budkin   ; S. A. Shakhova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 1118-1131

Résumé

Let $p$ be a prime number. Denote by $\mathfrak{R}_{\delta, \lambda}$ the non-abelian variety of nilpotent groups of class at most 2 of exponent $p^\delta$ with commutator subgroup of exponent $p^\lambda;$ by $F_2$ the free group of rank 2 in $\mathfrak{R}_{\delta, \lambda};$ by $qH$ the quasivariety of groups generated by a group $H.$ It is proved that the interval $[qF_2, qG]$ is continual if all the following conditions are true: $G\in\mathfrak{R}_{\delta, \lambda},$ $G$ is a finite group defined in $\mathfrak{R}_{\delta, \lambda}$ by commutator defining relations, $qF_2\varsubsetneq qG.$

Détail

Keywords: lattice, quasivariety, nilpotent group.

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