Levi classes of quasivarieties of nilpotent groups of class at most two

S. A. Shakhova

Algebra i logika, Tome 62 (2023) no. 6, pp. 742-761.

Résumé

A Levi class $L(\mathcal{M})$ generated by a class $\mathcal{M}$ of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to $\mathcal{M}$. Let $p$ be a prime and $p\neq 2$, let $H_{p}$ be a free group of rank $2$ in the variety of nilpotent groups of class at most $2$ with commutator subgroup of exponent $p$, and let $qH_{p}$ be the quasivariety generated by the group $H_{p}$. It is shown that there exists a set of quasivarieties $\mathcal{M}$ of cardinality continuum such that $L(\mathcal{M})=L(qH_{p})$. Let $s$ be a natural number, $s\geq 2$. We specify a system of quasi-identities defining $L(q(H_{p}, Z_{p^{s}}))$, and prove that there exists a set of quasivarieties $\mathcal{M}$ of cardinality continuum such that $L(\mathcal{M})=L(q(H_{p}, Z_{p^{s}}))$, where $Z_{p^{s}}$ is a cyclic group of order $p^{s}$; $q(H_{p}, Z_{p^{s}})$ is the quasivariety generated by the groups $H_{p}$ and $Z_{p^{s}}$.

Détail

Keywords: quasivariety, Levi class, nilpotent group.

Bibliographie

[1] F. W. Levi, “Groups in which the commutator operation satisfies certain algebraic conditions”, J. Indian Math. Soc., New Ser., 6 (1942), 87–97 | MR | Zbl

[2] L. C. Kappe, “On Levi-formations”, Arch. Math., 23:6 (1972), 561–572 | DOI | MR | Zbl

[3] L.-C. Kappe, R. F. Morse, “Groups with 3-Abelian normal closures”, Arch. Math., 51:2 (1988), 104–110 | DOI | MR | Zbl

[4] A. I. Budkin, “Kvazimnogoobraziya Levi”, Sib. matem. zh., 40:2 (1999), 266–270 | MR | Zbl

[5] A. I. Budkin, “O klassakh Levi, porozhdennykh nilpotentnymi gruppami”, Algebra i logika, 39:6 (2000), 635–647 | MR | Zbl

[6] A. I. Budkin, L. V. Taranina, “O kvazimnogoobraziyakh Levi, porozhdennykh nilpotentnymi gruppami”, Sib. matem. zh., 41:2 (2000), 270–277 | MR | Zbl

[7] R.F. Morse, “Levi-properties generated by varieties”, The mathematical legacy of Wilhelm Magnus. Groups, geometry and special functions, Conf. on the legacy of Wilhelm Magnus (May 1-3, 1992, Polytechnic Univ. Brooklyn, NY, USA), Contemp. Math., 169, eds. W. Abikoff et al., Am. Math. Soc., Providence, RI, 1994, 467–474 | DOI | MR | Zbl

[8] A. I. Budkin, V. A. Gorbunov, “K teorii kvazimnogoobrazii algebraicheskikh sistem”, Algebra i logika, 14:2 (1975), 123–142 | MR | Zbl

[9] V. V. Lodeischikova, “O kvazimnogoobraziyakh Levi, porozhdennykh nilpotentnymi gruppami”, Izv. Alt. gos. un-ta, 2009, no. 1(61), 26–29

[10] V. V. Lodeischikova, “O klassakh Levi, porozhdennykh nilpotentnymi gruppami”, Sib. matem. zh., 51:6 (2010), 1359–1366 | MR

[11] V. V. Lodeischikova, “O kvazimnogoobraziyakh Levi eksponenty $p^s$”, Algebra i logika, 50:1 (2011), 26–41 | MR

[12] S. A. Shakhova, “Ob aksiomaticheskom range klassov Levi”, Algebra i logika, 57:5 (2018), 587–600 | MR | Zbl

[13] S. A. Shakhova, “The axiomatic rank of the Levi class generated by the almost Abelian quasivariety of nilpotent groups”, Lobachevskii J. Math., 41:9 (2020), 1680–1683 | DOI | MR | Zbl

[14] S. A. Shakhova, “Klassy Levi kvazimnogoobrazii grupp s kommutantom eksponenty $p$”, Algebra i logika, 60:5 (2021), 510–524 | MR | Zbl

[15] V. V. Lodeischikova, S. A. Shakhova, “Klassy Levi kvazimnogoobrazii nilpotentnykh grupp eksponenty $p^{s}$”, Algebra i logika, 61:1 (2022), 77–92

[16] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1977 | MR

[17] A. I. Maltsev, Algebraicheskie sistemy, Nauka, M., 1970 | MR

[18] A. I. Budkin, Kvazimnogoobraziya grupp, Izd-vo Alt. gos. un-ta, Barnaul, 2002

[19] V. A. Gorbunov, Algebraicheskaya teoriya kvazimnogoobrazii, Sibirskaya shkola algebry i logiki, Nauch. kniga, Novosibirsk, 1999

[20] Kh. Neiman, Mnogoobraziya grupp, Mir, M., 1969

[21] S. A. Shakhova, “On the lattice of quasivarieties of nilpotent groups of class 2”, Sib. Adv. Math., 7:3 (1997), 98–125 | MR | Zbl

[22] A. N. Fedorov, Kvazitozhdestva konechnykh 2-nilpotentnykh grupp, dep. v VINITI, No 5489-V87, 1987