The Product Separability of the Generalized Free Product of Cyclic Groups

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 H 1 , H 2 , …, H n are finitely generated subgroups of G , then the product subset H 1 H 2 … 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 ]).