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Group Laws Implying Virtual Nilpotence

Published online by Cambridge University Press:  09 April 2009

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.

Information

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2003

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