• 한국수학사학회지
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• 제12권2호
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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.

• 대한수학회논문집
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• 제28권1호
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• pp.55-62
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• 2013

In [5], Johnson introduced the primitivity of subgroups and proved that a finite group G is supersolvable if every primitive subgroup of G has a prime power index in G. In that paper, he also posed an interesting problem: what a group looks like if all of its primitive subgroups are maximal. In this note, we give the detail structure of such groups in solvable case. Finally, we use the primitivity of some subgroups to characterize T-group and the solvable $PST_0$-groups.

• 대한수학회보
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• 제56권2호
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• pp.501-504
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• 2019

In this note, we prove that Gluck's conjecture, Isaacs-Navarro-Wolf Conjecture and Taketa's inequality are true for solvable groups whose character graphs having diameter 3.

• 대한수학회보
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• 제57권6호
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• pp.1475-1479
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• 2020

Let G be a finite group and ${\sigma}_1(G)={\frac{1}{{\mid}G{\mid}}}\;{\sum}_{H{\leq}G}\;{\mid}H{\mid}$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of G, we prove that if ${\sigma}_1(G)<{\frac{117}{20}}$, then G is solvable. This partially solves an open problem posed in [9].

• 대한수학회보
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• 제30권2호
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• pp.285-293
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• 1993

In [4], Baumslag and Tretkoff proved a residual finiteness criterion for HNN extensions (Theorem 1.2, below). This result has been used extensively in the study of the residual finiteness of HNN extensions. Note that every one-relator group can be embedded in a one-relator group whose relator has zero exponent sum on a generator, and the latter group can be considered as an HNN extension. Hence the properties of an HNN extension play an important role in the study of one-relator groups [3], [2]. In this paper we prove a criterion for HNN extensions to be .pi.$_{c}$(Theorem 2.2). Moreover, we can prove that certain one-relator groups, known to be residually finite, are actually .pi.$_{c}$. It was known by Mostowski [10] that the word problem is solvable for finitely presented, residually finite groups. In the same way, the power problem is solvable for finitely presented .pi.$_{c}$ groups. Another application of subgroup separability with respect to special subgroups was mentioned by Thurston [12, Problem 15].m 15].

• 대한수학회논문집
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• 제34권1호
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• pp.137-143
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• 2019

In this paper, we show that $A_4$ is the only finite group with exactly two conjugacy classes of the same size having some non-real linear characters.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.217-227
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• 2005

Let G be a finite group and $\#$Cent(G) denote the number of centralizers of its elements. G is called n-centralizer if $\#$Cent(G) = n, and primitive n-centralizer if $\#$Cent(G) = $\#$Cent($\frac{G}{Z(G)}$) = n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and if G is a finite group such that G/Z(G)$\simeq$$A_5$, then $\#$Cent(G) = 22 or 32. Moreover, we prove that As is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of As in terms of the number of centralizers

• 대한수학회보
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• 제48권6호
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• pp.1147-1155
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• 2011

In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.167-173
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• 2023

Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ is isomorphic to the dihedral group of order 2p where Z_∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

• Kyungpook Mathematical Journal
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• 제49권3호
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• pp.419-424
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• 2009

Let J be the Jacobian variety of a hyperelliptic curve over $\mathbb{Q}$. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free $\mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\widetilde{M}$ is an extension of M which contains all the torsion points of J over $\widetilde{\mathbb{Q}}$, then $J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $\widetilde{M}^{sol}$ is the maximal solvable extension of $\widetilde{M}$.