• Title/Summary/Keyword: free-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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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.

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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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.

REVISIT TO CONNECTED ALEXANDER QUANDLES OF SMALL ORDERS VIA FIXED POINT FREE AUTOMORPHISMS OF FINITE ABELIAN GROUPS

  • Sim, Hyo-Seob;Song, Hyun-Jong
    • East Asian mathematical journal
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    • v.30 no.3
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    • pp.293-302
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    • 2014
  • In this paper we provide a rigorous proof for the fact that there are exactly 8 connected Alexander quandles of order $2^5$ by combining properties of fixed point free automorphisms of finite abelian 2-groups and the classification of conjugacy classes of GL(5, 2). Furthermore we verify that six of the eight associated Alexander modules are simple, whereas the other two are semisimple.

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.

FREE ACTIONS ON THE NILMANIFOLD

  • Shin, Joonkook
    • Journal of the Chungcheong Mathematical Society
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    • v.10 no.1
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    • pp.161-175
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    • 1997
  • We classify free actions of finite abelian groups on the 3-dimensional nilmanifold, up to topological conjugacy. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal subgroups of almost Bieberbach groups of finite index, up to affine conjugacy.

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FREE ACTIONS OF FINITE ABELIAN GROUPS ON 3-DIMENSIONAL NILMANIFOLDS

  • Choi, Dong-Soon;Shin, Joon-Kook
    • Journal of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.795-826
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    • 2005
  • We study free actions of finite abelian groups on 3­dimensional nilmanifolds. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy. All such actions are completely classified.

CLASSIFICATION OF FREE ACTIONS OF FINITE GROUPS ON 3-DIMENSIONAL NILMANIFOLDS

  • Koo, Daehwan;Oh, Myungsung;Shin, Joonkook
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1411-1440
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    • 2017
  • We study free actions of finite groups on 3-dimensional nil-manifolds with the first homology ${\mathbb{Z}}^2{\oplus}{\mathbb{Z}}_p$. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy.

MULTIPLICITY-FREE ACTIONS OF THE ALTERNATING GROUPS

  • Balmaceda, Jose Maria P.
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.453-467
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    • 1997
  • A transitive permutation representation of a group G is said to be multiplicity-free if all of its irreducible constituents are distinct. The character corresponding to the action is called the permutation character, given by $(1_H)^G$, where H is the stabilizer of a point. Multiplicity-free permutation characters are of interest in the study of centralizer algebras and distance-transitive graphs, and all finite simple groups are known to have such characters. In this article, we extend to the alternating groups the result of J. Saxl who determined the multiplicity-free permutation representations of the symmetric groups. We classify all subgroups H for which $(1_H)^An, n > 18$, is multiplicity-free.

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