THE PRODUCT SEPARABILITY OF THE GENERALIZED FREE PRODUCT OF CYCLIC GROUPS

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 ], H_n are finitely generated subgroups of G, then the product subset H₁H₂ [ctdot ] H_n 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]).