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A note on Engel groups and local nilpotence

Published online by Cambridge University Press:  09 April 2009

Yuri Medvedev
Affiliation:
Department of Mathematics Statistics York University North York Ontario, M3J 1P3 Canada e-mail: rburns@mathstat.yorku.ca medvedev@mathstat. yorku.ca
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Abstract

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This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

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