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

Abstract

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 {p} for some prime number p. 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.1991 Mathematics Subject Classification. 20E06, 20E26 (primary).