Boatman, Nicholas S.; Olshanskii, Alexander Yu. On identities in the products of group varieties. (English) Zbl 1331.20036 Int. J. Algebra Comput. 25, No. 3, 531-540 (2015). 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)]. Cited in 1 Document 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 Keywords:group identities; varieties of groups; products of varieties; Burnside varieties; periodic groups; infinite \(p\)-groups; non-finitely based varieties; axiomatic ranks; basis ranks; bases of identities Citations:Zbl 1032.20020; Zbl 0125.01402 PDFBibTeX XMLCite \textit{N. S. Boatman} and \textit{A. Yu. Olshanskii}, Int. J. Algebra Comput. 25, No. 3, 531--540 (2015; Zbl 1331.20036) Full Text: DOI arXiv References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.