Abelian Subgroup Separability of Some Free, Product of Groups with Commuting Subgroups

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发表日期：

2025年2月1日

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N. Temate Tangang,
G. Mantika,
D. Tieudjo

摘要

Abstract

In this paper, we show that under some conditions, the free product of two finitely generated abelian subgroup separable groups with commuting subgroup is finitely generated abelian subgroup separable. Also, given \(\Gamma=(V(\Gamma),E(\Gamma))\) a finite simplicial graph and \(\mathcal{G}=\{G_{v}|v\in V(\Gamma)\}\) a collection of groups, we prove that if each of the groups \(G_{v}\) is abelian subgroup separable, then so is the graph product of groups \(G=\Gamma\mathcal{G}\) . Finally, since any abelian subgroup of a free group is finitely generated, we prove that every abelian subgroup of free product of two free groups with commuting abelian subgroups is finitely generated.