E. Sokolov

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

(Lobachevskii Journal of Mathematics, Vol.11, pp.27-38)

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 $\Sigma $ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\pi $@-groups.\linebreak 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 ^{\prime} $@-isolated, don't belong to $\Sigma $. Some sufficient conditions are obtained for $\Sigma $ to coincide with the family of all other $\pi ^{\prime} $@-isolated cyclic subgroups of group $G$. /BR It is proved, in particular, that the residual $p$@-finiteness of a free product with cyclic amalgamation implies the $p$@-separability of all $p^{\prime} $@-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$@-finite nilpotent groups



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