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. 
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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