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A note on Engel groups and local nilpotence

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

Published online by Cambridge University Press:  09 April 2009

R. G. Burns  and
Yuri Medvedev
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.

MSC classification

Secondary: 20F45: Engel conditions 20F19: Generalizations of solvable and nilpotent groups 20E10: Quasivarieties and varieties of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 1 , February 1998 , pp. 92 - 100
Copyright
Copyright © Australian Mathematical Society 1998

References

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