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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: [email protected]
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

References

[1]Black, S., ‘Which words spell “almost nilpotent” ?’, preprint, 1997.Google Scholar
[2]Burns, R. G., Macedońska, O. and Medvedev, Y., ‘Groups satisfying semigroup laws, and nilpotentby-Burnside varieties’, J. Algebra 195 (1997), 510–525.CrossRefGoogle Scholar
[3]Burns, R. G. and Medvedev, Y., ‘A note on Engel groups and local nilpotence’, J. Austral. Math. Soc. Ser. A 64 (1998), 92–100.CrossRefGoogle Scholar
[4]Groves, J. R. J., ‘Varieties of soluble groups and a dichotomy of P. Hall’, Bull. Austral. Math. Soc. 5 (1971), 391–410.CrossRefGoogle Scholar
[5]Gruenberg, K. W., ‘Two theorems on Engel groups’, Math. Proc. Cambridge Philos. Soc. 49 (1953), 377–380.CrossRefGoogle Scholar
[6]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[7]Jones, G. A., ‘Varieties and simple groups’, J. Austral. Math. Soc. 17 (1974), 163–173.CrossRefGoogle Scholar
[8]Kargapolov, M. I. and Merzljakov, Ju. I., Fundamentals of the theory of groups (Springer, Berlin, 1979).CrossRefGoogle Scholar
[9]Kim, Y. K. and Rhemtulla, A. H., ‘Weak maximality condition and polycyclic groups’, Proc. Amer Math. Soc. 123 (1995), 711–714.CrossRefGoogle Scholar
[10]Lubotzky, A. and Mann, A., ‘Powerful p-groups, I. Finite groups’, J. Algebra 105 (1987), 484–505.CrossRefGoogle Scholar
[11]Mal'cev, A. I., ‘Nilpotent semigroups’, Uchen. Zap. Ivanovsk. Ped. Inst. 4 (1953), 107–111.Google Scholar
[12]Neumann, B. H. and Taylor, T., ‘Subsemigroups of nilpotent groups’, Proc. Roy. Soc. Ser. A 274 (1963), 1–4.Google Scholar
[13]Neumann, H., Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[14]Olshanskii, A. Yu. and Storozhev, A., ‘A group variety defined by a semigroup law’, J. Austral. Math. Soc. Ser. A 60 (1996), 255–259.CrossRefGoogle Scholar
[15]Point, F., ‘Milnor identities’, Comm. Algebra 24 (1996), 3725–3744.CrossRefGoogle Scholar
[16]Robinson, D. J. S., Finiteness conditions and generalized soluble groups. Part I (Springer, Berlin, 1972).CrossRefGoogle Scholar
[17]Robinson, D. J. S., A course in the theory of groups (Springer, Berlin, 1982).CrossRefGoogle Scholar
[18]Rosenblatt, J. M., ‘Invariant measures and growth conditions’, Trans. Amer Math. Soc. 193 (1974), 33–53.CrossRefGoogle Scholar
[19]Semple, J. F. and Shalev, A., ‘Combinatorial conditions in residually finite groups, I’, J. Algebra 157 (1993), 43–50.CrossRefGoogle Scholar
[20]Shalev, A., ‘Combinatorial conditions in residually finite groups, II’, J. ALgebra 157 (1993), 51–62.CrossRefGoogle Scholar
[21]Zelmanov, E. I., ‘Engel Lie algebras’, Dokl. Akad. Nauk SSSR 292 (1987), 265–268.Google Scholar
[22]Zelmanov, E. I., ‘On some problems of group theory and Lie algebras’, Math. USSR-Sbornik 66 (1990), 159–168.CrossRefGoogle Scholar
[23]Zelmanov, E. I., ‘The solution of the restricted Burnside problem for 2-groups’, Mat. Sb. 182 (1991), 568–592.Google Scholar
[24]Zelmanov, E. I., ‘The solution of the restricted Burnside problem for groups of odd exponent’, Math. USSR-Izv. 36 (1991), 41–60.CrossRefGoogle Scholar