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Abelian subgroup separability of some one-relator groups. (English) Zbl 1091.20021
A subgroup $$M$$ of a group $$G$$ is called finitely separable if for any element $$g\in G$$, not belonging to $$M$$, there exists a homomorphism $$\varphi$$ of $$G$$ onto some finite group $$X$$ such that $$g\varphi\not\in M\varphi$$. If every cyclic subgroup of $$G$$ is finitely separable, then $$G$$ is said to be cyclic subgroup separable.
The author proves that any group given by the presentation $$\langle a,b;\;[a^n,b^m]=1\rangle$$, where $$m$$ and $$n$$ are integers greater than 1, is cyclic subgroup separable. The author establishes suitable properties of these groups which enable him to prove that every finitely generated Abelian subgroup of any such group is finitely separable.

##### MSC:
 20E26 Residual properties and generalizations; residually finite groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05 Generators, relations, and presentations of groups
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