• 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.

• 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.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.

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.203-219
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• 2007

We estimate the stable rank and connected stable rank of group $C^*$-algebra of certain disconnected solvable Lie groups such as semi-direct products of connected solvable Lie groups by the integers.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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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].

• 제어로봇시스템학회:학술대회논문집
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• 1992.10b
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• pp.405-412
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• 1992

The problem we consider in this paper is more demanding than the problem of input-output linearization with state equivalence recently solved by Cheng, Isidori, Respondek, and Tarn. We request that the MIMO nonlinear system, for which the problem of input-output linearization with state-equivalence is solvable, can be decoupled. In exchange for further requirement like this, our problem produces more usable and informative results than the problem of input-output linearization with state-equivalence. We present the necessary and sufficient conditions for our problem to be solvable. We characterize each of the nonlinear systems satisfying these conditions by a set of parameters which are invariant under the group action of state feedback and transformation. Using this set of parameters, we can determine directly the unique one, among the canonical forms of decouplable and controllable linear systems, to which a nonlinear system can be transformed via appropriate state feedback and transformation. Finally, we present the necessary and sufficient conditions for our problem to be solvable with internal stability, that is, for stable decoupling.

• The Mathematical Education
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• v.21 no.3
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• pp.15-17
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• 1983