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JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2024 Volume 21, Issue 2, Pages 1118–1131 (Mi semr1736)

Mathematical logic, algebra and number theory

On the complexity of the lattice of quasivarieties of nilpotent groups

A. I. Budkin, S. A. Shakhova

Altai State University, pr. Lenina, 61, 656049, Barnaul, Russia

Abstract: 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, $qF_2\varsubsetneq qG.$

Keywords: lattice, quasivariety, nilpotent group.

UDC: 512.54.01

MSC: 20F05

Received January 29, 2024, published November 25, 2024

DOI: 10.33048/semi.2024.21.073



© Steklov Math. Inst. of RAS, 2026