# On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

## (*Lobachevskii Journal of Mathematics, Volume XI*)

Let *G* be a free product of two groups with amalgamated subgroup, *π* be either the set of all prime numbers or the one-element set {*π*} for some prime number . Denote by *Σ* the family of all cyclic subgroups of group *G*, which are separable in the class of all finite *π*-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite *π*-index of group *G*, the subgroups conjugated with them and all subgroups, which aren't *π'*-isolated, don't belong to *Σ*. Some sufficient conditions are obtained for *Σ* to coincide with the family of all other *π'*-isolated cyclic subgroups of group *G*. It is proved, in particular, that the residual *p*-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.