E. Sokolov

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.
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