Algebraic Analysis of Ҟ‐Ϙ Group Operations on Neutrosophic Complex Fuzzy Subgroups and Their Normal Counterparts

This paper introduces a novel framework called the Ҟ‐Ϙ group acting on neutrosophic‐complex fuzzy subsets, expanding classical group theory to incorporate fuzziness, uncertainty, and indeterminacy. We formulate the algebraic structure and key properties of complex fuzzy subgroups within this neutrosophic environment, examining how closure, associativity, identity, and inverse elements operate when influenced by degrees of truth, falsity, and indeterminacy. The work further integrates the concept of homomorphisms by exploring how these complex fuzzy subgroups behave under structure‐preserving mappings. In particular, we investigate the homomorphic images and preimages of such subgroups and establish fundamental results that confirm the stability of their structural characteristics under transformations. Additionally, we develop the notion of direct products for the Ҟ‐Ϙ group operating neutrosophic‐complex fuzzy subgroups. Our findings show that the direct product construction retains the essential features of the underlying neutrosophic‐complex fuzzy subgroup, and this property extends naturally from the product of two subgroups to finite collections of them. The proposed framework enhances the interaction between fuzzy algebra and neutrosophic logic, offering a broader and more flexible mathematical model for analyzing complex group structures under uncertainty.