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The Product Separability of the Generalized Free Product of Cyclic Groups
Author(s) -
You Shihong
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610796004425
Subject(s) - separable space , mathematics , profinite group , free product , product (mathematics) , group (periodic table) , combinatorics , direct product , hausdorff space , cyclic group , finitely generated abelian group , topology (electrical circuits) , pure mathematics , discrete mathematics , mathematical analysis , physics , abelian group , geometry , quantum mechanics
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 ]).