• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1753-1763
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• 2013

In this paper, we show that tree products of certain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.61-71
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• 2007

A group G is called cyclic subgroup separable for the cyclic subgroup H if for each $x\;{\in}\;G{\backslash}H$, there exists a normal subgroup N of finite index in G such that $x\;{\not\in}\;HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-by-finite groups.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1375-1390
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• 2023

In this note, we shall show that the generalized free products of subgroup separable groups amalgamating a subgroup which itself is a finite extension of a finitely generated normal subgroup of both the factor groups are weakly potent and cyclic subgroup separable. Then we apply our result to generalized free products of finite extensions of finitely generated torsion-free nilpotent groups. Finally, we shall show that their tree products are cyclic subgroup separable.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.461-494
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• 2002

Let G = E${\ast}_{A}F$, where A is a finitely generated abelian subgroup. We prove a criterion for G to be {A}-double coset separable. Applying this result, we show that polygonal products of central subgroup separable groups, amalgamating trivial intersecting central subgroups, are double coset separable relative to certain central subgroups of their vertex groups. Finally we show that such polygonal products are conjugacy separable. It follows that polygonal products of polycyclic-by-finite groups, amalgamating trivial intersecting central subgroups, are conjugacy separable.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.813-828
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• 2013

It is known that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroups are cyclic or central in both factor groups. However, those generalized free products may not be conjugacy separable when the amalgamated subgroup is a direct product of two infinite cycles. In this paper we show that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroup is ${\langle}h{\rangle}{\times}D$, where D is in the center of both factors.

• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.913-920
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• 2004

We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its containing vertex group.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.521-530
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• 1997

We first prove a criterion for the conjugacy separability of free products with amalgamation where the amalgamated subgroup is not necessarily cyclic. Applying this result, we show that free products of finite number of polycyclic-by-finite groups with central amalgamation are conjugacy separable. We also show that polygonal products of polycyclic-by-finite groups, amalgamating central cyclic subgroups with trivial intersections, are conjugacy separable.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1741-1752
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• 2008

Grossman [7] showed that certain cyclically pinched 1-relator groups have residually finite outer automorphism groups. In this paper we prove that tree products of finitely generated free groups amalgamating maximal cyclic subgroups have residually finite outer automorphism groups. We also prove that polygonal products of finitely generated central subgroup separable groups amalgamating trivial intersecting central subgroups have residually finite outer automorphism groups.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1195-1204
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• 2010

In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non cyclic amalgamated subgroup. Applying this criterion, we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.77-85
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• 2001

If an Azumaya algebra A is a homomorphic image of a finite group ring RG where G is a direct product of subgroups then A can be decomposed into subalgebras A(sub)i which are homomorphic images of subgroup rings of RG. This result is extended to projective Schur algebras, and in this case behaviors of 2-cocycles will play major role. Moreover considering the situation that A is represented by Azumaya group ring RG, we study relationships between the representing groups for A and A(sub)i.