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A solution of a problem of Plotkin and Vovsi and an application to varieties of groups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

C. K. Gupta  and
A. N. Krasil'nikov
C. K. Gupta
Affiliation:
Department of Mathematics University of ManitobaWinnipeg R3T 2N2, Canada
A. N. Krasil'nikov
Affiliation:
Department of Algebra Moscow Pedagogical State University14 Krasnoprudnaya St. Moscow 107140, Russia
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Abstract

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Let K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

Keywords

relatively free groupsfully invariant subgroupsgroup algebrasfully invariant idealsfinitely based varieties of groups

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 3 , December 1999 , pp. 329 - 355
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bryant, R. M. and Newman, M. F., ‘Some finitely based varieties of groups’, Proc. London Math. Soc. (3) 28 (1974), 237–252.CrossRefGoogle Scholar
[2]Cohen, D. E., ‘On the laws of a metabelian variety’, J. Algebra 5 (1967), 267–273.CrossRefGoogle Scholar
[3]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
[4]Higman, G., ‘Ordering by divisibility in abstract algebras’, Proc. London Math. Soc. (3) 2 (1952), 326–336.CrossRefGoogle Scholar
[5]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;Google Scholar
English translation: Math. USSR Izv. 37 (1991), 539–553.CrossRefGoogle Scholar
[6]Neumann, H., Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[7]Newman, M. F., ‘Just non-finitely-based varieties of groups’, Bull. Austral. Math. Soc. 4 (1971), 343–348.CrossRefGoogle Scholar
[8]Plotkin, B. I. and Vovsi, S. M., Varieties of group representations: General theory, connections and applications (Russian) (Zinatne, Riga, 1983).Google Scholar
[9]Popov, A. P., ‘Some finitely based varieties of rings’, C. R. Acad. Bulgare Sci. 32 (1979), 855–858.Google Scholar
[10]Powell, M. B.Oates, S., ‘Identical relations in finite groups’, J. Algebra 1 (1964), 11–39.Google Scholar
[11]Vaughan-Lee, M. R., ‘Uncountably many varieties of groups’, Bull. London Math. Soc. 2 (1970), 280–286.CrossRefGoogle Scholar
[12]Vovsi, S. M., Topics in varieties of group representations, London Math. Soc. Lecture Note Ser. 163 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar