• The Pure and Applied Mathematics
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• v.15 no.4
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• pp.353-364
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• 2008

The purpose of this paper is to investigate the finite extension of weighted word L-delta groups. The paper revealed that a finite extension of a weighted word L-delta group is a weighted word L-delta group, and an abelian group, in addition, is a weighted word L-delta group and simultaneously a word L-delta group.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.951-960
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• 2022

In this paper, we determine the finite p-group such that the intersection of its any two distinct minimal nonabelian subgroups is a maximal subgroup of the two minimal nonabelian subgroups, and the finite p-group in which any two distinct 𝓐₁-subgroups generate an 𝓐₂-subgroup. As a byproduct, we answer a problem proposed by Berkovich and Janko.

• Journal for History of Mathematics
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• v.13 no.2
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• pp.151-160
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• 2000

In this paper we prove that for all irreducible complex characters of a finite group, there exist F-representations and a finite degree field extension F ⊇ Q, where Q is the rational number field.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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• pp.479-487
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• 2004

Let G be a finite group and N be a normal subgroup of G. We denote by ncc(N) the number of conjugacy classes of N in G and N is called n-decomposable, if ncc(N) = n. Set $K_{G}\;=\;\{ncc(N)$\mid$N{\lhd}G\}$. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X. In this paper we characterise the {1, 3, 4}-decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9), the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n{2{\frac{n-1}{2}}\;-\;1)$, in which n is odd positive integer and $2{\frac{n-1}{2}}\;-\;1$ is a Mersenne prime or a metabelian group of order $2^n(2{\frac{n}{3}}\;-\;1)$, where 3$\mid$n and $2\frac{n}{3}\;-\;1$ is a Mersenne prime. Moreover, we calculate the set $K_{G}$, for some finite group G.

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.81-96
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• 2015

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if $H{\cap}K{\neq}1$ where 1 denotes the trivial subgroup of G. In this paper we characterize all finite groups whose intersection graphs are planar. Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2, 10, 11, 12].

• 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.

• Journal of the Korean Mathematical Society
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• v.23 no.2
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• pp.201-215
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• 1986

• Journal of the Korean Mathematical Society
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• v.17 no.1
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• pp.91-97
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• 1980

• Journal for History of Mathematics
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• v.12 no.2
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• pp.143-149
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• 1999

Every finite solvable group is only a subgroup of an M-groups and all M-groups are solvable. Supersolvable group is an M-groups and also subgroups of solvable or supersolvable groups are solvable or supersolvable. But a subgroup of an M-groups need not be an M-groups . It has been studied that whether a normal subgroup or Hall subgroup of an M-groups is an M-groups or not. In this note, we investigate some historical research background on the M-groups and also we give some conditions that a normal subgroup of an M-groups is an M-groups and show that a solvable group is an M-group.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.83-116
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• 2020

Let G be a finite group. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. The large Davenport constant D(G) is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length D(G) over dihedral and dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.