Abstract
Let \( {\mathcal{R}}_{p^{k}} \) be the variety of \( 2 \)-nilpotent groups of exponent \( p^{k} \) with commutator subgroup of exponent \( p \) (\( p \) is a prime). We prove the infinity of the set of the subquasivarieties of \( {\mathcal{R}}_{p^{k}} \) \( (k\geq 2) \) generated by a finite group and lacking any independent bases of quasi-identities.
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References
Budkin A. I., “Independent axiomatizability of quasivarieties of groups,” Math. Notes, vol. 31, no. 6, 817–826 (1982).
Budkin A. I., “Independent axiomatizability of quasivarieties of generalized solvable groups,” Algebra and Logic, vol. 25, no. 3, 155–166 (1986).
Medvedev N. Ya., “On quasivarieties of \( Z \)-groups and groups,” Sib. Math. J., vol. 26, no. 5, 717–723 (1985).
Budkin A. I., “Quasivarieties of groups having no coverings,” Math. Notes, vol. 37, no. 5, 333–337 (1985).
Fedorov A. N., “Quasi-identities of a free 2-nilpotent group,” Math. Notes, vol. 40, no. 5, 837–841 (1986).
Fedorov A. N., “Subquasivarieties of nilpotent minimal non-Abelian group varieties,” Sib. Math. J., vol. 21, no. 6, 840–850 (1980).
Shakhova S. A., “On the lattice of quasivarieties of nilpotent groups of class 2,” Siberian Adv. Math., vol. 7, no. 3, 98–125 (1997).
Kravchenko A. V., Nurakunov A. M., and Schwidefsky M. V., “Structure of quasivariety lattices. I. Independent axiomatizability,” Algebra and Logic, vol. 57, no. 6, 445–462 (2018).
Kravchenko A. V., Nurakunov A. M., and Schwidefsky M. V., “Structure of quasivariety lattices. II. Undecidable problems,” Algebra and Logic, vol. 58, no. 2, 123–136 (2019).
Adams M. E. and Dziobiak W., “\( Q \)-Universal quasivarieties of algebras,” Proc. Amer. Math. Soc., vol. 120, no. 4, 1053–1059 (1994).
Sapir M. V., “The lattice of quasivarieties of semigroups,” Algebra Universalis, vol. 21, no. 2–3, 172–180 (1985).
Shakhova S. A., “On the quasivariety generated by a finite \( p \)-group,” Math. Notes, vol. 53, no. 1, 345–347 (1993).
Shakhova S. A., “On a cardinality of the lattice of quasivarieties of nilpotent groups,” Algebra and Logic, vol. 38, no. 3, 202–206 (1999).
Budkin A. I., “A lattice of quasivarieties of nilpotent groups,” Algebra and Logic, vol. 33, no. 1, 14–21 (1994).
Budkin A. I., “Quasivarieties with torsion-free nilpotent groups,” Algebra and Logic, vol. 40, no. 6, 353–364 (2001).
Budkin A. I., Quasivarieties of Groups [Russian], Altai University, Barnaul (2002).
Gorbunov V. A., Algebraic Theory of Quasivarieties, Plenum, New York (1998).
Malcev A. I., Algebraic Systems, Springer and Akademie, Berlin, Heidelberg, and New York (1973).
Fedorov A. N., “Quasivarieties of finite 2-nilpotent groups,” submitted to VINITI, no. 5489–B87 (1987).
Olshanskii A. Yu., “Conditional identities in finite groups,” Sib. Math. J., vol. 15, no. 6, 1000–1003 (1974).
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Budkin, A.I. On the Quasivarieties Generated by a Finite Group and Lacking Any Independent Bases of Quasi-Identities. Sib Math J 61, 983–993 (2020). https://doi.org/10.1134/S0037446620060038
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DOI: https://doi.org/10.1134/S0037446620060038