We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups.

Keywords:

HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite

MSC Classifications:

20E26, 20E06, 20F10 show english descriptions Residual properties and generalizations; residually finite groups
Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 20E26 - Residual properties and generalizations; residually finite groups
20E06 - Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F10 - Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70]

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