CHARACTERIZATION OF NORMAL FUZZY SUBGROUP - Bagas Dwi Cahyo; Suroto Suroto; Sri Maryani (Universitas Wiralodra, 2025)
In this article, we discuss the characterization of a normal fuzzy subgroup of classical group . The discussion of this characterization is carried out using Abelian properties, fuzzy conjugate subgroups, fuzzy normalizers, ????-level sets, and fuzzy cosets. The result shows that a sufficient and necessary condition for a normal fuzzy subgroup is the fulfilment of the Abelian condition in the fuzzy subgroup. Then, the equality between of the membership value of all element of and its conjugate elements is also a sufficient and necessary condition for normal fuzzy subgroup of ????. Moreover, sufficient and necessary conditions of the normal fuzzy subgroup are the normalizer of this fuzzy subgroup is equal to ????. Henceforth, the sufficient and necessary conditions of a normal fuzzy subgroup of is its ????-level set is a normal subgroup of . Meanwhile, the similarity of the fuzzy right coset and fuzzy left coset of the fuzzy subgroup is also a sufficient and necessary condition for the normal fuzzy subgroup. Furthermore, the normal properties of subgroups on classical groups are a special case of the normal properties in fuzzy subgroups.