On Jordan-Holder Theorem under Intuitionistic Fuzzy Groups

Abstract

The theory of intuitionistic fuzzy groups is an algebraic structure derivable from the utilization of groups in intuitionistic fuzzy sets. Many notions in group theory have been presented in intuitionistic fuzzy group theory. However, concepts like simple group, maximal normal subgroup, normal series, composition series, and the Jordan-Holder Theorem are open problems in intuitionistic fuzzy group theory. Hence, this paper defines simple intuitionistic fuzzy groups, maximal normal intuitionistic fuzzy subgroups, normal series for intuitionistic fuzzy groups, and composition series for intuitionistic fuzzy groups with illustrations. In addition, the Jordan-Holder Theorem in intuitionistic fuzzy group theory is verified. It is shown that every intuitionistic fuzzy group of a finite group possesses a composition series, and any two composition series for an intuitionistic fuzzy group of a finite group are equivalent.