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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 [email protected]
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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

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