Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T06:43:44.105Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on Engel groups and local 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
Yuri Medvedev
Affiliation:
Department of Mathematics Statistics York University North York Ontario, M3J 1P3 Canada e-mail: [email protected] medvedev@mathstat. yorku.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Burns, R. G., Macedonska, Olga and Medvedev, Yuri, ‘Groups satisfying semigroup laws, and nilpotent-by-Burnside varieties’, J. Algebra 195 (1997), 510–525.CrossRefGoogle Scholar
[2]Groves, J. R. J., ‘Varieties of soluble groups and a dichotomy of P. Hall’, Bull. Austral. Math. Soc. 5 (1971), 391–410.CrossRefGoogle Scholar
[3]Gruenberg, K. W., ‘Two theorems on Engel groups’, Proc. Camb. Phil. Soc. 49 (1953), 377–380.CrossRefGoogle Scholar
[4]Gruenberg, K. W., ‘The upper central series in soluble groups’, Illinois J. Math. 5 (1961), 436–466.CrossRefGoogle Scholar
[5]Gupta, N. D. and Newman, M. F., ‘Third Engel groups’, Bull. Austral. Math. Soc. 40 (1989), 215–230.CrossRefGoogle Scholar
[6]Heineken, H., ‘Engelsche Elemente der Länge drei’, Illinois J. Math. 5 (1961), 681–707.CrossRefGoogle Scholar
[7]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[8]Kargapolov, M. I. and Merzljakov, Ju. I., Fundamentals of the theory of groups (Springer, Berlin, 1979).CrossRefGoogle Scholar
[9]Mal'cev, A. I., ‘Nilpotent semigroups’, Uchen. Zap. Ivanovsk. Ped. Inst. 4 (1953), 107–111.Google Scholar
[10]Robinson, D. J. S., A course in the theory of groups (Springer, Berlin, 1982).CrossRefGoogle Scholar
[11]Wilson, J. S., ‘Two-generator conditions in residually finite groups’, Bull. London Math. Soc. 23 (1991), 239–248.CrossRefGoogle Scholar
[12]Wilson, J. S. and Zelmanov, E. I., ‘Identities for Lie algebras of pro-p groups’, J. Pure Appl. Algebra 81 (1992), 103–109.CrossRefGoogle Scholar