• Title, Summary, Keyword: abelian groups

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NON-ABELIAN TENSOR ANALOGUES OF 2-AUTO ENGEL GROUPS

  • MOGHADDAM, MOHAMMAD REZA R.;SADEGHIFARD, MOHAMMAD JAVAD
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1097-1105
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    • 2015
  • The concept of tensor analogues of right 2-Engel elements in groups were defined and studied by Biddle and Kappe [1] and Moravec [9]. Using the automorphisms of a given group G, we introduce the notion of tensor analogue of 2-auto Engel elements in G and investigate their properties. Also the concept of $2_{\otimes}$-auto Engel groups is introduced and we prove that if G is a $2_{\otimes}$-auto Engel group, then $G{\otimes}$ Aut(G) is abelian. Finally, we construct a non-abelian 2-auto-Engel group G so that its non-abelian tensor product by Aut(G) is abelian.

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.

ON ALMOST ω1-pω+n-PROJECTIVE ABELIAN p-GROUPS

  • Danchev, Peter V.
    • Korean Journal of Mathematics
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    • v.22 no.3
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    • pp.501-516
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    • 2014
  • We define the class of almost ${\omega}_1-p^{\omega+n}$-projective abelian p-primary groups and investigate their basic properties. The established results extend classical achievements due to Hill (Comment. Math. Univ. Carol., 1995), Hill-Ullery (Czech. Math. J., 1996) and Keef (J. Alg. Numb. Th. Acad., 2010).

ON ALMOST n-SIMPLY PRESENTED ABELIAN p-GROUPS

  • Danchev, Peter V.
    • Korean Journal of Mathematics
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    • v.21 no.4
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    • pp.401-419
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    • 2013
  • Let $n{\geq}0$ be an arbitrary integer. We define the class of almost n-simply presented abelian p-groups. It naturally strengthens all the notions of almost simply presented groups introduced by Hill and Ullery in Czechoslovak Math. J. (1996), n-simply presented p-groups defined by the present author and Keef in Houston J. Math. (2012), and almost ${\omega}_1-p^{{\omega}+n}$-projective groups developed by the same author in an upcoming publication [3]. Some comprehensive characterizations of the new concept are established such as Nunke-esque results as well as results on direct summands and ${\omega}_1$-bijections.

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 $0{\longrightarrow}B{\longrightarrow}Res_{L/K}(A){\longrightarrow}C{\longrightarrow}0$.

ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS

  • Yu, Hoseog
    • Honam Mathematical Journal
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    • v.37 no.1
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    • pp.1-6
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    • 2015
  • Let A be an abelian variety defined over a number field K. Let L be a biquadratic extension of K with Galois group G and let III (A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming III(A/L) is finite, we compute [III(A/K)]/[III(A/L)] where [X] is the order of a finite abelian group X.

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.

A CLASSIFICATION OF PRIME-VALENT REGULAR CAYLEY MAPS ON ABELIAN, DIHEDRAL AND DICYCLIC GROUPS

  • Kim, Dong-Seok;Kwon, Young-Soo;Lee, Jae-Un
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.17-27
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    • 2010
  • A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.