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Group Laws Implying Virtual Nilpotence

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

R. G. Burns  and
Yuri Medvedev
R. G. Burns
Affiliation:
Department of Mathematics and Statistics York UniversityToronto, Ontario Canada e-mail: rbums@pascal.math.yorku.ca
Yuri Medvedev
Affiliation:
Bank of Montreal Toronto, Ontario Canada M3J 1P3
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Abstract

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If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups 20E10: Quasivarieties and varieties of groups 20F45: Engel conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 295 - 312
Copyright
Copyright © Australian Mathematical Society 2003

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