Geodesic
On the cyclic subgroup separability of free products of two groups with amalgamated subgroup
Lobachevskii journal of mathematics, Tome 11 (2002), pp. 27-38.

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Let $G$ be a free product of two groups with amalgamated subgroup, $\pi$ be either the set of all prime numbers or the one-element set $\{p\}$ for some prime number $p$. Denote by $\sum$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\pi$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $\pi$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $\pi'$-isolated, don't belong to $\sum$. Some sufficient conditions are obtained for $\sum$ to coincide with the family of all other $\pi'$-isolated cyclic subgroups of group $G$. It is proved, in particular, that the residual $\pi'$-finiteness of a free product with cyclic amalgamation implies the $p$-separability of all $p'$-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$-finite nilpotent groups.
Keywords: Generalized free products, cyclic subgroup separability. 
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E. V. Sokolov. On the cyclic subgroup separability of free products of two groups with amalgamated subgroup. Lobachevskii journal of mathematics, Tome 11 (2002), pp. 27-38. http://geodesic.mathdoc.fr/item/LJM_2002_11_a5/

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