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THE PRODUCT SEPARABILITY OF THE GENERALIZED FREE PRODUCT OF CYCLIC GROUPS

Published online by Cambridge University Press:  01 August 1997

SHIHONG YOU
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
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Abstract

Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, whenever H1, H2, [ctdot ], Hn are finitely generated subgroups of G, then the product subset H1H2 [ctdot ] Hn is closed in G. In this paper, we prove that if G=F×Z is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).

Type
Research Article
Copyright
The London Mathematical Society 1997

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