• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.497-503
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• 2012

In this paper, we study elliptic curves E over $\mathbb{Q}$ such that the 3-torsion subgroup E[3] is split as ${\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$. For a non-zero intege $m$, let $C_m$ denote the curve $x^3+y^3=m$. We consider the relation between the set of integral points of $C_m$ and the elliptic curves E with $E[3]{\simeq}{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$.

• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.279-288
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• 2020

Niebrzydowski and Przytycki conjectured that the torsion of rack and quandle homology of a dihedral quandle of order 2k is annihilated by k, unless k = 2^t for t > 1. We partially prove this conjecture.

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.285-292
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• 1998

We introduce the notion of TL-p-subgroups that is an ex-tension of the notion of fuzzy p=subgroups and show that a torsion TL-subgroup of an Abelian group with T=${\bigwedge}$ can be written as the intersection of its minimal TL-p-subgroups.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1375-1390
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• 2023

In this note, we shall show that the generalized free products of subgroup separable groups amalgamating a subgroup which itself is a finite extension of a finitely generated normal subgroup of both the factor groups are weakly potent and cyclic subgroup separable. Then we apply our result to generalized free products of finite extensions of finitely generated torsion-free nilpotent groups. Finally, we shall show that their tree products are cyclic subgroup separable.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.87-95
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• 2019

The homothety classes of lattices in a two dimensional vector space over a nonarchimedean local field form a regular tree ${\mathcal{T}}$ of degree q + 1 on which the projective linear group acts naturally where q is the order of the residue field. We show that for any finite regular combinatorial graph of even degree q + 1, there exists a torsion free discrete subgroup ${\Gamma}$ of the projective linear group such that ${\mathcal{T}}/{\Gamma}$ is isomorphic to the graph.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.157-161
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• 2007

Let G be a p-mixed abelian group with semi-complete torsion subgroup $G_t$ such that G is splitting or is of torsion-free rank one, and let R be a commutative unitary ring of prime characteristic p. It is proved that the group algebras RG and RH are R-isomorphic for any group H if and only if G and H are isomorphic. This isomorphism relationship extends our earlier results in (Southeast Asian Bull. Math., 2002), (Proc. Amer. Math. Soc., 2002) and (Bull. Korean Math. Soc., 2005) as well as completely settles a problem posed by W. May in (Proc. Amer. Math. Soc., 1979).

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1998.03a
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• pp.3-6
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• 1998

We introduce the notion of TL-p-subgroups that is an extension of the notion of fuzzy p-subgroups and show that a torsion TL-subgroup of an Abelian group with T=∧ can be written as the intersection of its minimal TL-p-subgroups.

• Proceedings of the Korea Inteligent Information System Society Conference
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• 2000.04a
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• pp.475-478
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• 2000

Recently encryption algorithms based on difficulties of factorization have been used with popularization. Prime number factorizations are progressed rapidly. In this paper, characteristics of elliptic curve are analyzed and generation of elliptic curves suitable for prime number factorization is discussed.

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.225-239
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• 2003

(G, <) is an ordered group if'<'is a total order relation on G in which f < g implies that xfy < xgy for all f, g, x, y $\in$ G. We say that (G, <) is centrally ordered if (G, <) is ordered and ［G,D］ $\subseteq$ C for every convex jump C $\prec$ D in G. Equivalently, if $f^{-1}g f{\leq} g^2$ for all f, g $\in$ G with g > 1. Every order on a torsion-free locally nilpotent group is central. We prove that if every order on every two-generator subgroup of a locally soluble orderable group G is central, then G is locally nilpotent. We also provide an example of a non-nilpotent two-generator metabelian orderable group in which all orders are central.