A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup

This research presents an innovative concept known as the Neutrosophic group operating on fuzzy subsets, extending the traditional algebraic framework to include elements of fuzziness and neutrosophy. The study defines the structure and algebraic properties of Neutrosophic group operating fuzzy subgroups, highlighting how these subsets behave under group operations influenced by uncertainty and indeterminacy. Several foundational algebraic characteristics are discussed, including closure, associativity, identity, and inverses within this specialized fuzzy context. In addition, the notion of homomorphisms is incorporated by examining how fuzzy subgroups within the Neutrosophic group behave under structure-preserving mappings. Specifically, we analyze the homomorphic image and preimage of these subgroups, establishing essential results that contribute to understanding their structural consistency and transformation properties. Furthermore, the concept of the direct product of Neutrosophic group operating fuzzy subgroups is introduced. We demonstrate that the direct product maintains the integrity of the fuzzy subgroup structure, preserving its key features across multiple components. This idea is not only applied to the direct product of two such subgroups but is also extended to a finite number of them, reinforcing the robustness and generalizability of the model. These developments open new avenues in fuzzy algebra and Neutrosophic logic-based group theory.