• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.83-93
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• 2017

The Cayley graph ${\Gamma}=Cay(G,S)$ is called normal edge-transitive if $N_A(R(G))$ acts transitively on the set of edges of ${\Gamma}$, where $A=Aut({\Gamma})$ and R(G) is the regular subgroup of A. In this paper, we determine all hexavalent normal edge-transitive Cayley graphs on groups of order pqr, where p > q > r > 2 are prime numbers.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.14 no.2
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• pp.154-161
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• 2014

We discuss some interesting sublattices of interval-valued fuzzy subgroups. In our main result, we consider the set of all interval-valued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We then prove that this set forms a modular sublattice of the lattice of interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we employ a lattice diagram to exhibit the interrelationship among these sublattices.

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.103-113
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• 1991

Some properties of a path space and the fundamental group ${\sigma}(X,x_0,G)$ of a topological transformation group (X, G, ${\pi}$) are described. It is shown that ${\sigma}(X,x_0,H)$ is a normal subgroup of ${\sigma}(X,x_0,G)$ if H is a normal subgroup of G ; Let (X, G, ${\pi}$) be a transformation group with the open action property. If every identification map $p:{\Sigma}(X,x,G)\;{\longrightarrow}\;{\sigma}(X,x,G)$ is open for each $x{\in}X$, then ${\lambda}$ induces a homeomorphism between the fundamental groups ${\sigma}(X,x_0,G)$ and ${\sigma}(X,y_0,G)$ where ${\lambda}$ is a path from $x_0$ to $y_0$ in X ; The space ${\sigma}(X,x_0,G)$ is an H-space if the identification map $p:{\Sigma}(X,x_0,G)\;{\longrightarrow}\;{\sigma}(X,x_0,G)$ is open in a topological transformation group (X, G, ${\pi}$).

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.371-376
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• 2008

Let $P(M,G,{\pi})=:P$ be a principal fibre bundle with structure Lie group G over a base manifold M. In this paper we get the following facts: 1. The tangent bundle TG of the structure Lie group G in $P(M,G,{\pi})=:P$ is a Lie group. 2. The Lie algebra ${\mathcal{g}}=T_eG$ is a normal subgroup of the Lie group TG. 3. $TP(TM,TG,{\pi}_*)=:TP$ is a principal fibre bundle with structure Lie group TG and projection ${\pi}_*$ over base manifold TM, where ${\pi}_*$ is the differential map of the projection ${\pi}$ of P onto M. 4. for a Lie group $H,\;TH=H{\circ}T_eH=T_eH{\circ}H=TH$ and $H{\cap}T_eH=\{e\}$, but H is not a normal subgroup of the group TH in general.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.945-959
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• 2008

n-ary $H_v$-structures is a generalisation of both n-ary structures and $H_v$-structures. A wide class of n-ary $P-H_v$-groups is the n-ary $P-H_v$-groups that is concidered in this paper. In this paper the notion of a normal subgroup of an n-ary $P-H_v$-groups is introduced and the isomorphism theorems for n-ary $P-H_v$-groups are stated and proved. Also some examples and related properties are investigated.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.3
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• pp.231-244
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• 2013

We introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: First, for any interval-valued fuzzy congruence $R_e$ on a group G, the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G. Second, for any interval-valued fuzzy congruence R on a groupoid S, we show that a binary operation * an S=R is well-defined and also we obtain some results related to additional conditions for S. Also we improve that for any two interval-valued fuzzy congruences R and Q on a semigroup S such that $R{\subset}Q$, there exists a unique semigroup homomorphism g : S/R${\rightarrow}$S/G.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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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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• v.63 no.2
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• pp.187-198
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• 2023

The motivation for this study comes from a question posed by I.N. Herstein in the Israel Journal of Mathematics in 1978. Specifically, let D be a division ring with center F. The aim of this paper is to demonstrate that every quasinormal subgroup of the multiplicative group of D, which is radical over some proper division subring, is central if one of the following conditions holds: (i) D is weakly locally finite; (ii) F is uncountable; or (iii) D is the Mal'cev-Neumann division ring.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.785-796
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• 2024

Let p be a prime. A group G is said to be residually p-finite if for each non-trivial element x of G, there exists a normal subgroup N of index a power of p in G such that x is not in N. In this note we shall prove that certain HNN extensions of free abelian groups of finite rank are residually p-finite. In addition some of these HNN extensions are subgroup separable. Characterisations for certain one-relator groups and similar groups including the Baumslag-Solitar groups to be residually p-finite are proved.

• Journal of Gastric Cancer
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• v.18 no.3
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• pp.274-286
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• 2018

Purpose: The aim of this study was to investigate the impact of preoperative low body mass index (BMI) on both the short- and long-term outcomes in patients with gastric cancer. Materials and Methods: A total of 510 patients with gastric cancer were divided into the following 3 groups: low BMI group (${\leq}18.5kg/m^2$, n=51), normal BMI group ($18.6-24.9kg/m^2$, n=308), and high BMI group (${\geq}25.0kg/m^2$, n=151). Results: There were significantly more stage III/IV patients in the low BMI group than in the other groups (P=0.001). Severe postoperative complications were more frequent (P=0.010) and the survival was worse (P<0.001) in the low BMI group. The subgroup analysis indicated that survival was worse in the low BMI group of the stage I/II subgroup (P=0.008). The severe postoperative complication rate was higher in the low BMI group of the stage III/IV subgroup (P=0.001), although the recurrence rate and survival did not differ in the stage III/IV subgroup among all the BMI groups. Low BMI was an independent poor prognostic factor in the stage I/II subgroup (disease-free survival: hazard ratio [HR], 13.521; 95% confidence interval [CI], 1.186-154.197; P=0.036 and overall survival: HR, 5.130; 95% CI, 1.644-16.010; P=0.005), whereas low BMI was an independent risk factor for severe postoperative complications in the stage III/IV subgroup (HR, 17.158; 95% CI, 1.383-212.940; P=0.027). Conclusions: Preoperative low BMI in patients with gastric cancer adversely affects survival among those with stage I/II disease and increases the severe postoperative complication rate among those with stage III/IV disease.