Research Article On Fundamental Algebraic Characterizations of μ-Fuzzy Normal Subgroups Ibtisam Masmali , 1 Umer Shuaib , 2 Abdul Razaq , 3 Areeba Fatima, 2 and Ghaliah Alhamzi 4 1 Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia 2 Department of Mathematics, Government College University, Faisalabad, Pakistan 3 Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan 4 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia Correspondence should be addressed to Umer Shuaib; mumershuaib@gcuf.edu.pk Received 12 March 2022; Revised 17 April 2022; Accepted 30 April 2022; Published 19 May 2022 Academic Editor: Mohammed S. Abdo Copyright © 2022 Ibtisam Masmali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, we present the study of μ-fuzzy subgroups and prove numerous fundamental algebraic attributes of this newly defined notion. We also define the concept of μ-fuzzy normal subgroup and investigate many vital algebraic characteristics of these phenomena. In addition, we characterize the quotient group induced by this particular fuzzy normal subgroup and establish a group isomorphism between the quotient groups G/κ μ and G/κ μ ∗ . Furthermore, we initiate the study of level subgroup, open level subgroup, and tangible subgroup of a μ-fuzzy subgroup and emphasize the significance of μ-fuzzy normal subgroups by establishing a relationship between these newly defined notions and μ-fuzzy normal subgroup. 1. Introduction The theory of fuzzy logic is based on the concept of relative graded membership as inspired by the processes of human perception. This logic deals with information that is uncer- tain, imprecise, vague, partly true, or without clear bound- aries. Moreover, this theory provides a mathematical framework within which ambiguous conceptual phenomena can be studied with precision. New computing methods based on fuzzy logic can be used in the development of intel- ligent system for decision-making, identification, pattern recognition, optimization, and control system. This particu- lar logic is currently being used in the industrial practice of advanced information technology. One of the most impor- tant applications of group theory is its key role in geometry and cryptography. Geometry is the study of properties of a space that are invariant under a group of transformations of that space. The theory of groups is used to classify the symmetries of molecules, crystal structures, and regular polyhedral. It is also used to solve the old issues of algebra. Fuzzy subsets (FSs) have a central position in modern mathematics. In 1965, Zadeh [1] proposed the idea of FSs. The idea of fuzzy subgroup (FSG) was presented by Rosen- feld [2] in the framework of FSs in 1971. Das [3] defined the level subgroups (LSGs) of a FSG in 1981. Mukherjee and Bhattacharya [4] characterized the notions of fuzzy nor- mal subgroup (FNSG) and fuzzy coset in 1984. Mashour et al. [5] described many important properties of FNSGs in 1990. The characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups were presented in [6]. To see more on the development of theory of FSGs, we refer to [7–11]. The fuzzy logic was effec- tively used in industrial management [12], coding theory [13] and forecasting systems [14]. Rowlands and Wang [15] applied fuzzy logic to present a useful technique to design fuzzy-SPC evaluation and control method in 2000. Hindawi Journal of Function Spaces Volume 2022, Article ID 2703489, 10 pages https://doi.org/10.1155/2022/2703489