• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2141-2147
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• 2017

Let G be a finite group and m a divisor of ${\mid}G{\mid}$. We prove that G has at least ${\tau}(m)$ cyclic subgroups whose orders divide m, where ${\tau}(m)$ is the number of divisors of m.

• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.913-920
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• 2004

We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its containing vertex group.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1753-1763
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• 2013

In this paper, we show that tree products of certain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.525-540
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• 2013

We introduce the concept of level subgroups of an interval-valued fuzzy subgroup and study some of its properties. These level subgroups in turn play an important role in the characterization of all interval-valued fuzzy subgroup of a prime cyclic group.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.6 no.3
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• pp.240-246
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• 2006

We introduce the concept of level subgroups of an intuitionistic fuzzy subgroup and study some of it's properties. These level subgroups in turn play an important role in the characterization of all intuitionistic fuzzy subgroup of a prime cyclic group.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.479-486
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• 2016

The norm N(G) of a group G is the intersection of the normalizers of all the subgroups of G. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which asserts that a finite group G is cyclic if and only if Aut(G)/P(G) is cyclic, where P(G) is the power automorphism group of G.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1049-1067
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• 2019

In this paper, we discuss the number of distinct fuzzy subgroups of the group ${\mathbb{Z}}_{p^n}{\times}{\mathbb{Z}}_{q^m}{\times}{\mathbb{Z}}_r$, m = 1, 2, 3 where p, q, r are distinct primes for any $n{\in}{\mathbb{Z}}^+$ using the criss-cut method that was proposed by Murali and Makamba in their study of distinct fuzzy subgroups. The criss-cut method first establishes all the maximal chains of the subgroups of a group G and then counts the distinct fuzzy subgroups contributed by each chain. In this paper, all the formulae for calculating the number of these distinct fuzzy subgroups are given in polynomial form.

• 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.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2004.04a
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• pp.389-393
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• 2004

In this paper, we introduce the concept of level subgroups of an intuitionistic fuzzy subgroup, and study some properties of level subgroups in the first part of the paper. These level subgroups in turn play an important role in the characterization of all intuitionistic fuzzy subgroups of a prime cyclic group.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.367-376
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• 2015

In this paper, we classified finite p-groups G such that $$C_G(x)/<x>$$ is cyclic for all non-central elements $x{\in}G$. This solved a problem proposed By Y. Berkovoch.