Incorporating £-Complex Intuitionistic Fuzzy Set to Sylow Theorems in Group Theory

The complex intuitionistic fuzzy (CIF) set is an advanced version of the regular intuitionistic fuzzy set. It is made to better show the uncertainty and complexity that arise in real-life problems. The grading and nongrading degrees in the CIF set are shown by complex-valued functions that are defined on the unit disc of the complex plane. A CIF set combines both magnitude and phase terms to establish a more robust mathematical framework to understand and address, such as decision-making and pattern detection, where conventional intuitionistic fuzzy sets may be insufficient. The £-CIF set is the generalization of the CIF set. The extension of the traditional Sylow theorems in the context of £-CIF brings in the idea of the £-CIF conjugate element within the £-CIF subgroup of a group. In this paper, we introduce the paradigmatic idea of £-CIF subgroups of a group, and various features of this concept are demonstrated. Furthermore, the construction of the £-CIF version of the Cauchy theorem is derived. Finally, we look into the £-CIF Sylow p subgroup for a finite group and show how Sylow’s theorems can be extended in a £-CIF environment.