• Communications of the Korean Mathematical Society
• /
• v.26 no.3
• /
• pp.349-371
• /
• 2011

Using the "belongs to" relation and "quasi-coincident with" relation between a fuzzy point and a fuzzy subgroup, Bhakat and Das, in 1992 and 1996, initiated general types of fuzzy subgroups which are a generalization of Rosenfeld's fuzzy subgroups. In this paper, more general notions of "belongs to" and "quasi-coincident with" relation between a fuzzy point and a fuzzy set are provided, and more general formulations of general types of fuzzy (normal) subgroups by Bhakat and Das are discussed. Furthermore, general type of coset is introduced, and related fundamental properties are investigated.

• Communications of the Korean Mathematical Society
• /
• v.16 no.1
• /
• pp.75-83
• /
• 2001

The various fuzzy subgroups of a group which are admissible under operator domains are studied. In particular, the classes of all inner automorphisms, automorphisms, and endomorphisms are applied on the fuzzy subgroups of a group. As results, several theorems and examples concerning the fuzzy subgroups following from these kinds of operator domains are obtained. Moreover, we prove that a necessary condition for a fuzzy subgroup to be characteristic is that the center of the fuzzy subgroup is characteristic.

• Journal of the Korean Mathematical Society
• /
• v.50 no.3
• /
• pp.579-589
• /
• 2013

A finite group G is called an NSN-group if every proper subgroup of G is either normal in G or self-normalizing. In this paper, the non-NSN-groups whose proper subgroups are all NSN-groups are determined.

• Journal of Korean Society for Quality Management
• /
• v.25 no.3
• /
• pp.74-85
• /
• 1997

In this paper we calculate the subgroup size necessary for detecting the change of percent defective with several detection probabilities for orginal population fraction nonconforming p, changed population fraction nonconforming $p^*$, and the ratio k=$p^*$/p in the usage of p control charts. From our calculation we can know the error level of normal a, pp.oximation in detection probability calculation and recommend the subgroup size with lower error levels of normal a, pp.oximation, and then we show the reasonable subgroup size necessary for p, $p^*$, k, and the detection probability of the change of fraction nonconforming in a process. The information that we here show in tables will be useful when p control chart users decide the subgroup size in the p control chart users decide the subgroup size in the p control chart.

• Korean Journal of Mathematics
• /
• v.26 no.3
• /
• pp.373-385
• /
• 2018

In this paper we introduce the notion of a fuzzy kernel of a fuzzy homomorphism on groups and we show that it is a fuzzy normal subgroup of the domain group. Conversely, we also prove that any fuzzy normal subgroup is a fuzzy kernel of some fuzzy epimorphism, namely the canonical fuzzy epimorphism. Finally, we formulate and prove the fuzzy version of the fundamental theorem of homomorphism and those isomorphism theorems.

• Communications of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.1087-1093
• /
• 2020

The aim of this paper is to show that every Engel quasinormal subgroups of the unit group of a division ring with uncountable center is central.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1835-1843
• /
• 2018

In this paper, finite p-groups G satisfying $N_G(H){\leq}H^G$ for every non-normal subgroup H of G are completely classified. This solves a problem proposed by Y. Berkovich.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.2
• /
• pp.445-449
• /
• 2013

We consider a congruence ${\rho}$ on a group G as a subsemigroup of the direct product $G{\times}G$. It is well known that a relation ${\rho}$ on G is a congruence if and only if there exists a normal subgroup N of G such that ${\rho}=\{(s,\;t):st^{-1}{\in}N\}$. In this paper we prove that if G is a finitely presented group, and if N is a normal subgroup of G with finite index, then the congruence ${\rho}=\{(s,\;t):st^{-1}{\in}N\}$ on G is finitely presented.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.5
• /
• pp.1375-1390
• /
• 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 applied mathematics & informatics
• /
• v.18 no.1_2
• /
• pp.657-663
• /
• 2005

In this paper, we consider groups of permutations S on a set A acting on subsets X of A. In particular, we show that if $X_2{\subseteq}X_1{\subseteq}A$ and Y is an S-normal extension of $X_2 in X_1$, then the Galois group $G_{S}(X_1/Y){\;}of{\;}X_1{\;}over{\;}X_2$ relative to S is an S-closed subgroup of $G_{S}(X_1/X_2)$ in the setting of a Galois theory (correspondence) for this situation.