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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

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