• Title, Summary, Keyword: polycyclic-by-finite groups

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CONJUGACY SEPARABILITY OF FREE PRODUCTS WITH AMALGAMATION

  • Kim, Goan-Su
    • 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.

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POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

  • Wong, Kok-Bin;Wong, Peng-Choon
    • 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.

CONJUGACY SEPARABILITY OF GENERALIZED FREE PRODUCTS OF FINITELY GENERATED NILPOTENT GROUPS

  • Zhou, Wei;Kim, Goan-Su;Shi, Wujie;Tang, C.Y.
    • 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.

OUTER AUTOMORPHISM GROUPS OF CERTAIN POLYGONAL PRODUCTS OF GROUPS

  • Kim, Goan-Su
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.1
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    • pp.45-52
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    • 2008
  • We show that certain polygonal products of any four groups, amalgamating central subgroups with trivial intersections, have Property E. Using this result, we derive that outer automorphism groups of polygonal products of four polycyclic-by-finite groups, amalgamating central subgroups with trivial intersections, are residually finite.

CONJUGACY SEPARABILITY OF CERTAIN FREE PRODUCT AMALGAMATING RETRACTS

  • Kim, Goan-Su
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.811-827
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    • 2000
  • We find some conditions to derive the conjugacy separability of the free product of conjugacy separable split extensions amalgamated along cyclic retracts. These conditions hold for any double coset separable groups and free-by-cyclic groups with nontrivial center. It was known that free-by-finite, polycyclic-by-finite, and fuchsian groups are double coset separable. Hence free products of those groups amalgamated along cyclic retracts are conjugacy separable.

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SEPARABILITY PROPERTIES OF CERTAIN POLYGONAL PRODUCTS OF GROUPS

  • Kim, Goan-Su;Tang, C.Y.
    • 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.

CYCLIC SUBGROUP SEPARABILITY OF CERTAIN GRAPH PRODUCTS OF SUBGROUP SEPARABLE GROUPS

  • Wong, Kok Bin;Wong, Peng Choon
    • 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.