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On identities in the products of group varieties. (English) Zbl 1331.20036

Summary: Let \(\mathcal B_n\) be the variety of groups satisfying the identity \(x^n=1\). It is proved that for every sufficiently large prime \(p\), say \(p>10^{10}\), the product \(\mathcal B_p\mathcal B_p\) cannot be defined by a finite set of identities. This solves the problem formulated by C. K. Gupta and A. Krasilnikov in 2003 [Ill. J. Math. 47, No. 1-2, 273-283 (2003; Zbl 1032.20020)]. We also find the axiomatic and the basis ranks of the variety \(\mathcal B_p\mathcal B_p\). For this goal, we improve the estimate for the basis rank of the product of group varieties obtained by G. Baumslag, B. H. Neumann, H. Neumann and P. M. Neumann in 1964 [Math. Z. 86, 93-122 (1964; Zbl 0125.01402)].

MSC:

20E10 Quasivarieties and varieties of groups
20F50 Periodic groups; locally finite groups
20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
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[1] Adyan S. I., Izv. Akad. Nauk SSSR, Ser. Math. 34 pp 721– (1970)
[2] Yu. A. Bahturin and A. Yu. Olshanskii, Algebra II, Encyclopaedia of Mathematical Sciences 18 (Springer-Heidelberg, 1991) pp. 107–221.
[3] DOI: 10.1007/BF01111576 · Zbl 0123.02202 · doi:10.1007/BF01111576
[4] DOI: 10.1007/BF01111331 · Zbl 0125.01402 · doi:10.1007/BF01111331
[5] DOI: 10.1017/S1446788700013914 · Zbl 0272.20019 · doi:10.1017/S1446788700013914
[6] DOI: 10.1016/0021-8693(67)90039-7 · Zbl 0157.34802 · doi:10.1016/0021-8693(67)90039-7
[7] Gupta C. K., Illinois J. Math. 47 pp 273– (2003)
[8] Hall M., The Theory of Groups (1959) · Zbl 0084.02202
[9] DOI: 10.1006/jabr.1996.6941 · Zbl 0918.20031 · doi:10.1006/jabr.1996.6941
[10] Kleiman Yu. G., Izv. Akad. Nauk SSSR, Ser. Math. 37 pp 475– (1973)
[11] Kleiman Yu. G., Izv. Akad. Nauk SSSR, Ser. Math. 38 pp 95– (1974)
[12] Krasilnikov A. N., Izv. Akad. Nauk SSSR, Ser. Math. 54 pp 1181– (1990)
[13] DOI: 10.1090/S0002-9939-1952-0049889-9 · doi:10.1090/S0002-9939-1952-0049889-9
[14] Lyndon R. C., Combinatorial Group Theory (2001) · Zbl 0997.20037 · doi:10.1007/978-3-642-61896-3
[15] DOI: 10.1007/BF01594191 · Zbl 0016.35102 · doi:10.1007/BF01594191
[16] DOI: 10.1007/978-3-642-88599-0 · doi:10.1007/978-3-642-88599-0
[17] DOI: 10.1016/0021-8693(64)90004-3 · Zbl 0121.27202 · doi:10.1016/0021-8693(64)90004-3
[18] Olshanskii A. Yu., Izv. Akad. Nauk SSSR, Ser. Math. 34 pp 376– (1970)
[19] A. Yu. Olshanskii, Geometry of Defining Relations in Groups (Nauka, Moscow, 1989) p. 448.
[20] DOI: 10.1007/978-3-662-07241-7 · doi:10.1007/978-3-662-07241-7
[21] Shirvanyan V. L., Izv. Akad. Nauk SSSR, Ser. Math. 40 pp 190– (1976)
[22] Shmelkin A. L., Izv. Akad. Nauk SSSR, Ser. Math. 29 pp 149– (1965)
[23] DOI: 10.4153/CJM-1956-031-x · Zbl 0075.01503 · doi:10.4153/CJM-1956-031-x
[24] DOI: 10.1112/blms/2.3.280 · Zbl 0216.08401 · doi:10.1112/blms/2.3.280
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