## 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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