• 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.31 no.2
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• pp.261-262
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• 2016

We present a short and very elementary proof that each finite BCK-algebra is solvable. It is a positive answer to the question posed in [4].

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.805-812
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• 2006

In this paper we find a suitable bound for the number of commutators which is required to express every element of the derived group of a solvable group satisfying the maximal condition for normal subgroups. The precise formulas for expressing every element of the derived group to the minimal number of commutators are given.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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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.

• Kyungpook Mathematical Journal
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• v.64 no.2
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• pp.337-352
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• 2024

In this paper, first we show that all finite BCI-algebras are solvable. We then introduce the notion of a θ-pair for a maximal ideal of a BCI-algebra. Proving various properties of maximal θ-pairs, we use them to characterize solvable and nilpotent BCI-algebras.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.471-478
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• 2006

In this paper we prove that if AX=b is a partial sign-solvable linear system with A being sign non-singular matrix and if ${\alpha}=\{j:\;x_j\;is\;sign-determined\;by\; Ax=b\}, then $A_{\alpha}X_{\alpha}=b_{\alpha}$ is a sign-solvable linear system, where $A_{\alpha}$ denotes the submatrix of A occupying rows and columns in o and xo and be are subvectors of x and b whose components lie in ${\alpha}$. For a sign non-singular matrix A, let $A_l,\;...,A_{\kappa}$ be the fully indecomposable components of A and let ${\alpha}_i$ denote the set of row numbers of $A_r,\;r=1,\;...,\;k$. We also show that if $A_x=b$ is a partial sign-solvable linear system, then, for $r=1,\;...,\;k$, if one of the components of xor is a fixed zero solution of Ax=b, then so are all the components of x_{{\alpha}r}$.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.867-873
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• 2024

A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra ℝD⋉n, where D is a derivation of n whose restriction to the center of n has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.363-366
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• 2015

In this note, we prove Gluck's conjecture and Isaacs-Navarro-Wolf Conjecture are true for the solvable groups with disconnected graphs by using Lewis' group structure theorem with respect to the disconnected character degree graphs.

• The Pure and Applied Mathematics
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• v.10 no.2
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• pp.103-118
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• 2003

We estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by topological Abelian groups. As an application we estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by solvable Lie groups. In particular, we obtain the stable rank estimate of twisted group $C^{*}-algebras$ of solvable Lie groups by the (reduced) dimension and (generalized) rank of groups.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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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.