Geodesic
Levi classes of quasivarieties of nilpotent groups of exponent $p^s$
Algebra i logika, Tome 61 (2022) no. 1, pp. 77-92.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $p=3$.
Keywords: quasivariety, Levi class, nilpotent group. 
@article{AL_2022_61_1_a3,
     author = {V. V. Lodeishchikova and S. A. Shakhova},
     title = {Levi classes of quasivarieties of nilpotent groups of exponent $p^s$},
     journal = {Algebra i logika},
     pages = {77--92},
     publisher = {mathdoc},
     volume = {61},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2022_61_1_a3/}
}
TY  - JOUR
AU  - V. V. Lodeishchikova
AU  - S. A. Shakhova
TI  - Levi classes of quasivarieties of nilpotent groups of exponent $p^s$
JO  - Algebra i logika
PY  - 2022
SP  - 77
EP  - 92
VL  - 61
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2022_61_1_a3/
LA  - ru
ID  - AL_2022_61_1_a3
ER  - 
%0 Journal Article
%A V. V. Lodeishchikova
%A S. A. Shakhova
%T Levi classes of quasivarieties of nilpotent groups of exponent $p^s$
%J Algebra i logika
%D 2022
%P 77-92
%V 61
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2022_61_1_a3/
%G ru
%F AL_2022_61_1_a3
V. V. Lodeishchikova; S. A. Shakhova. Levi classes of quasivarieties of nilpotent groups of exponent $p^s$. Algebra i logika, Tome 61 (2022) no. 1, pp. 77-92. http://geodesic.mathdoc.fr/item/AL_2022_61_1_a3/

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

[2] 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

[3] L. C. Kappe, W. P. Kappe, “On three-Engel groups”, Bull. Aust. Math. Soc., 7:3 (1972), 391–405

[4] K. W. Weston, “${\mathrm{ZA}}$-groups which satisfy the $m$-th Engel condition”, Ill. J. Math., 8 (1964), 458–472

[5] H. Heineken, “Engelsche Elemente der Länge drei”, Ill. J. Math., 5 (1961), 681–707

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

[7] A. I. Budkin, “Operator $L_{n}$ na kvazimnogoobraziyakh universalnykh algebr”, Sib. matem. zh., 60:4 (2019), 724–733

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

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

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

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

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

[13] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1977

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

[15] A. I. Maltsev, Algebraicheskie sistemy, Nauka, M., 1970

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

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

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

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

[20] A. N. Fedorov, “O podkvazimnogoobraziyakh nilpotentnykh minimalnykh ne abelevykh mnogoobrazii grupp”, Sib. matem. zh., 21:6 (1980), 117–131