• Honam Mathematical Journal
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• v.32 no.1
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• pp.45-51
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• 2010

Let A be an abelian variety defined over a number field K and let L be a cyclic extension of K with Galois group G = <${\sigma}$> of order n. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assume III(A/L) is finite. Let M(x) be a companion matrix of 1+x+${\cdots}$+$x^{n-1}$ and let $A^x$ be the twist of $A^{n-1}$ defined by $f^{-1}{\circ}f^{\sigma}$ = M(x) where $f:A^{n-1}{\rightarrow}A^x$ is an isomorphism defined over L. In this paper we compute [III(A/K)][III($A^x$/K)]/[III(A/L)] in terms of cohomology, where [X] is the order of an finite abelian group X.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.271-274
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• 1994

The concept of a continuous module is a generalization of that of an injective module, and conditions ($C_1$), (C$_2$) and ($C_3$) are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)

• Journal of the Korean Institute of Intelligent Systems
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• v.22 no.5
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• pp.646-655
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• 2012

First, we prove a number of results about interval-valued fuzzy groups involving the notions of interval-valued fuzzy cosets and interval-valued fuzzy normal subgroups which are analogs of important results from group theory. Also, we introduce analogs of some group-theoretic concepts such as characteristic subgroup, normalizer and abelian groups. Secondly, we prove that if A is an interval-valued fuzzy subgroup of a group G such that the index of A is the smallest prime dividing the order of G, then A is an interval-valued fuzzy normal subgroup. Finally, we show that there is a one-to-one correspondence the interval-valued fuzzy cosets of an interval-valued fuzzy subgroup A of a group G and the cosets of a certain subgroup H of G.

• Honam Mathematical Journal
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• v.27 no.3
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• pp.369-388
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• 2005

First, we prove a number of results about intuitionistic fuzzy groups involving the notions of intuitionistic fuzzy cosets and intuitionistic fuzzy normal subgroups which are analogs of important results from group theory. Also, we introduce analogs of some group-theoretic concepts such as characteristic subgroup, normalizer and Abelian groups. Secondly, we prove that if A is an intuitionistic fuzzy subgroup of a group G such that the index of A is the smallest prime dividing the order of G, then A is an intuitionistic fuzzy normal subgroup. Finally, we show that there is a one-to-one correspondence the intuitionistic fuzzy cosets of an intuitionistic fuzzy subgroup A of a group G and the cosets of a certain subgroup H of G.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1165-1178
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• 2009

In this paper we classify finite groups whose nonnormal subgroups are of order p or pq, where p, q are primes. As a by-product, we also classify the finite groups in which all nonnormal subgroups are cyclic.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.449-457
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• 2020

The aim of this short paper is to show some rigidity results for the actions of certain finitely presented groups by homeomorphisms. As an interesting and special case, we show that the actions of Higman-Thompson groups by homeomorphisms on a cohomology manifold with a non-zero Euler characteristic should be trivial. This is related to the wellknown Zimmer program and shows that the actions by homeomorphism could be very much different from those by diffeomorphisms.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.195-206
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• 2020

The fundamental idea of this article is to present an effective way to obtain the large groups in terms of the split extension obtained by a finite cyclic group and a free abelian group rank 2. The proof of the main result on largeness property of this specific split extension groups will be given by using the connection of large groups with the groups having deficiency one presentations.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.285-287
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• 2019

• Kyungpook Mathematical Journal
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• v.48 no.3
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• pp.495-502
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• 2008

We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the pro-p topology on each of the factors.

• 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.