• Title/Summary/Keyword: finite abelian groups

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FINITE p-GROUPS ALL OF WHOSE SUBGROUPS OF CLASS 2 ARE GENERATED BY TWO ELEMENTS

  • Li, Pujin;Zhang, Qinhai
    • Journal of the Korean Mathematical Society
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    • v.56 no.3
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    • pp.739-750
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    • 2019
  • We proved that finite p-groups in the title coincide with finite p-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite p-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.

RESIDUAL p-FINITENESS OF CERTAIN HNN EXTENSIONS OF FREE ABELIAN GROUPS OF FINITE RANK

  • Chiew Khiam Tang;Peng Choon Wong
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.785-796
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    • 2024
  • Let p be a prime. A group G is said to be residually p-finite if for each non-trivial element x of G, there exists a normal subgroup N of index a power of p in G such that x is not in N. In this note we shall prove that certain HNN extensions of free abelian groups of finite rank are residually p-finite. In addition some of these HNN extensions are subgroup separable. Characterisations for certain one-relator groups and similar groups including the Baumslag-Solitar groups to be residually p-finite are proved.

RELATIVE RELATION MODULES OF FINITE ELEMENTARY ABELIAN p-GROUPS

  • Yamin, Mohammad;Sharma, Poonam Kumar
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.4
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    • pp.1205-1210
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    • 2014
  • Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that $$E/S{\sim_=}G$$ is finite. For a prime p, $\hat{S}=S/S^{\prime}S^p$ may be regarded as an $F_pG$-module via conjugation in E. The aim of this article is to prove that $\hat{S}$ is decomposable into two indecomposable modules for finite elementary abelian p-groups G.

TATE-SHAFAREVICH GROUPS AND SCHANUEL'S LEMMA

  • Yu, Hoseog
    • Honam Mathematical Journal
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    • v.39 no.2
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    • pp.137-141
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    • 2017
  • Let A be an abelian variety defined over a number field K and let L be a finite Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Let $Res_{L/K}(A)$ be the restriction of scalars of A from L to K and let B be an abelian subvariety of $Res_{L/K}(A)$ defined over K. Assuming that III(A/L) is finite, we compute [III(B/K)][III(C/K)]/[III(A/L)], where [X] is the order of a finite abelian group X and the abelian variety C is defined by the exact sequence defined over K $0{\longrightarrow}B{\longrightarrow}Res_{L/K}(A){\longrightarrow}C{\longrightarrow}0$.

RESIDUAL FINITENESS AND ABELIAN SUBGROUP SEPARABILITY OF SOME HIGH DIMENSIONAL GRAPH MANIFOLDS

  • Kim, Raeyong
    • Korean Journal of Mathematics
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    • v.29 no.3
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    • pp.603-612
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    • 2021
  • We generalize 3-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally CAT(0). (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

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.

ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS

  • Yu, Hoseog
    • Honam Mathematical Journal
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    • v.38 no.1
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    • pp.85-93
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    • 2016
  • Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.