Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T03:50:45.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Metanilpotent varieties of groups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant  and
A. N. Krasil'nikov
R. M. Bryant
Affiliation:
UMIST, PO Box 88, Manchester M60 1QD, UK e-mail: bryant@umist.ac.uk
A. N. Krasil'nikov
Affiliation:
Moscow Pedagogical State University, 14 Krasnoprudnaya ul., Moscow 107140, Russia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For each positive integer n let N2, n denote the variety of all groups which are nilpotent of class at most 2 and which have exponent dividing n. For positive integers m and n, let N2, mN2, n denote the variety of all groups which have a normal subgroup in N2, m with factor group in N2, n. It is shown that if G ∈N2, mN2, n, where m and n are coprime, then G has a finite basis for its identities.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 1 , August 2002 , pp. 55 - 84
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Atkinson, M. D., ‘Alternating trilinear forms and groups of exponent 6’, J. Austral. Math. Soc. 16 (1973), 111–128.CrossRefGoogle Scholar
[2]Brady, J. M., Bryce, R. A. and Cossey, J., ‘On certain abelian-by-nilpotent varieties’, Bull. Austral. Math. Soc. 1 (1969), 403–416.CrossRefGoogle Scholar
[3]Bryant, R. M. and Newman, M. F., ‘Some finitely based varieties of groups’, Proc. London Math. Soc. (3) 28 (1974), 237–252.CrossRefGoogle Scholar
[4]Cohen, D. E., ‘On the laws of a metabelian variety’, J. Algebra 5 (1967), 267–273.CrossRefGoogle Scholar
[5]Gupta, C. K. and Krasil'nikov, A. N., ‘Some non-finitely based varieties of groups and group representations’, Internat. J. Algebra Comput. 5 (1995), 343–365.CrossRefGoogle Scholar
[6]Higman, G., ‘Ordering by divisibility in abstract algebras’, Proc. London Math. Soc. (3) 2 (1952), 326–336.CrossRefGoogle Scholar
[7]Higman, G., ‘Some remarks on varieties of groups’, Quart. J. Math. Oxford (2) 10 (1959), 165–178.CrossRefGoogle Scholar
[8]Higman, G., ‘The orders of relatively free groups’, in: Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ., 1965 (eds. Kovács, L. G. and Neumann, B. H.) (Gordon and Breach, New York, 1967) pp. 153–165.Google Scholar
[9]Kovács, L. G. and Newman, M. F., ‘Hanna Neumann's problems on varieties of groups’, in: Proc. Second Internat. Conf. Theory of Groups, Austral. Nat. Univ., 1973 (ed. Newman, M. F.), Lecture Notes in Math. 372 (Springer, Berlin, 1974) pp. 417–431.Google Scholar
[10]Krasil'nikov, A. N., ‘The identities of a group with nilpotent commutator subgroup are finitely based’, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 1181–1195; English translation: Math. USSR Izvestiya 37 (1991), 539–553.Google Scholar
[11]Lyndon, R. C., ‘Two notes on nilpotent groups’, Proc. Amer. Math. Soc. 3 (1952), 579–583.CrossRefGoogle Scholar
[12]McKay, S., ‘On centre-by-metabelian varieties of groups’, Proc. London Math. Soc. (3) 24 (1972), 243–256.CrossRefGoogle Scholar
[13]Neumann, H., Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[14]Shmel'kin, A. L., ‘Wreath products and varieties of groups’, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 149–170.Google Scholar