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On the cyclic subgroup separability of the free product of two groups with commuting subgroups. (English) Zbl 1308.20031
Author’s summary: Let $$G$$ be the free product of groups $$A$$ and $$B$$ with commuting subgroups $$H\leq A$$ and $$K\leq B$$, and let $$\mathcal C$$ be the class of all finite groups or the class of all finite $$p$$-groups. We derive the description of all $$\mathcal C$$-separable cyclic subgroups of $$G$$ provided this group is residually a $$\mathcal C$$-group. We prove, in particular, that if $$A,B$$ are finitely generated nilpotent groups and $$H,K$$ are $$p'$$-isolated in the free factors, then all $$p'$$-isolated cyclic subgroups of $$G$$ are separable in the class of all finite $$p$$-groups. The same statement is true provided $$A,B$$ are free and $$H,K$$ are $$p'$$-isolated and cyclic.

##### MSC:
 2e+27 Residual properties and generalizations; residually finite groups 2e+07 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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##### References:
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