Fuzzifying The Subgroups of The Cartesian Product of The Generalized Quaternion Group and The Cyclic Group

Abstract

This research investigates the number of distinct fuzzy subgroups in the Cartesian product of the generalized quaternion group and the cyclic group of order four. Building on prior foundational works and researches , this study extends the characterization of fuzzy subgroups within complex group structures. The generalized quaternion group, known for its non-commutative properties, combined with the cyclic group, creates a rich algebraic environment for examining subgroup behaviors. By employing both theoretical and computational methods, the research identifies and classifies the distinct fuzzy subgroups, revealing new insights into their structures, cardinality, and interrelations. The results are crucial for expanding the mathematical understanding of fuzzy group theory, especially within the realm of finite p-groups. In addition to providing a comprehensive count of the distinct fuzzy subgroups, this project contributes to the ongoing exploration of fuzzy subgroup structures in algebra, with potential applications in cryptography, group theory, and algorithmic mathematics. The findings offer significant progress in understanding the intersection of fuzzy logic and group theory, while also proposing new directions for future work in higher-order generalized quaternion and cyclic groups.