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Characterisation of nilpotent-by-finite groups

Published online by Cambridge University Press:  17 April 2009

Nadir Trabelsi
Nadir Trabelsi
Affiliation:
Département de Mathématiques, Université Ferhat Abbas, Sétif 19000, Algérie, e-mail maths@elhidhab.cerist.dz
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Abstract

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Let G be a finitely generated soluble group. The main result of this note is to prove that G is nilpotent-by-finite if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and two positive integers m = m (x,y), n = n (x, y) satisfying [x, nym] = 1. We prove also that if G is infinite and if m is a positive integer, then G is nilpotent-by-(finite of exponent dividing m) if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and a positive integer n = n (x,y) satisfying [x,nym] = 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 33 - 38
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Chapuis, O., ‘Variétés de groupes et m-identities’, C.R. Acad. Sci. Paris Ser. I 316 (1993), 15–17.Google Scholar
[2]Endimioni, G., ‘Groups covered by finitely many nilpotent subgroups’, Bull. Austral. Math.Soc. 50 (1994), 459–464.CrossRefGoogle Scholar
[3]Golod, E.S., ‘Some problems of Burnside type’, Amer. Math. Soc. Transl. Ser. 2 84 (1969), 83–88.Google Scholar
[4]Lennox, J.C., ‘Bigenetic properties of finitely generated hyper-(abelian-by-finite) groups’, J. Austral. Math. Soc. 16 (1973), 309–315.CrossRefGoogle Scholar
[5]Lennox, J.C. and Roseblade, J.E., ‘Centrality in finitely generated soluble groups’, J. Algebra 16 (1970), 399–435.CrossRefGoogle Scholar
[6]Lennox, J.C. and Wiegold, J., ‘Extensions of a problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 31 (1981), 459–463.CrossRefGoogle Scholar
[7]Longobardi, P., Maj, M. and Rhemtulla, A.H., ‘Infinite groups in a given variety and Ramsey's theorem’, Comm. in Algebra 20 (1992), 127–139.CrossRefGoogle Scholar
[8]Mal'cev, A. I., Collected works (Nauka, Moscow, 1976).Google Scholar
[9]Neumann, B.H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 21 (1976), 467–472.CrossRefGoogle Scholar
[10]Robinson, D.J.S., Finiteness conditions and generalized soluble groups (Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[11]Robinson, D.J.S., A course in the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[12]Zelmanov, E.I., “On some problems of group theory and Lie algebras’, Math. USSR-Sb 66 (1990), 159–168.CrossRefGoogle Scholar