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

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Premkumar M1*[0000-0002-8637-063X], Kiruthika K2, J. Saral2(a), D. J. Samatha Naidu3, Karuppusamy M3(a), Roselin Suhi R4, Usha P5 and Venkatachalam M6 1*Department of Mathematics, Sathyabama Institute of Sci ence and Technology (Deemed To Be University), Chennai-600119, Tamilnadu, India. Email: 1*[email protected] 2Department of Mathematics, K.S.Rangasamy College of Technology, Tiruchengode-637215, Tamilnadu, India, Email: 2[email protected] 2(a)Department of Mathematics, SRM Institute of Science and Technology, Ramapuram, Chennai, Tamilnadu, India. Email: 2(a)[email protected] 3Principal, Annamacharya PG College of Computer Studies, New Boyanapalli, Rajam-pet, Annamayya, Andhra Pradesh, India. Email: 3[email protected] 3(a)Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Erode-638401, Tamilnadu, India. Email: 3(a) [email protected] 4Department of Mathematics, Mar Ephraem College of Engineering Technology, Elavuvilai, Marthandam-629171, Tamil Nadu, India. Email: 4[email protected] 5Department of Science and Humanities, Karpagam Academy of Higher Education, Coimbatore, Tamilnadu, India. Email: 5ushaashrigmail.com 6Department of Mathematics, Erode Sengunthar Engineering College, Perundurai, Erode, Tamilnadu, India. Email: 6[email protected] *Correspondence: 1*[email protected] Abstract: 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. Neutrosophic Sets and Systems, Vol. 97, 2026 576 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Keywords: Fuzzy Set (FS), Neutrosophic group operating on fuzzy subsets , Neutrosophic group operating fuzzy subgroups , Neutrosophic group operating fuzzy normal subgroups . 1. Introduction In 1994, Ajmal [1] introduced a novel notation for group homomorphisms, the correspondence theorem, and fuzzy quotient groups. Mordeson created Fuzzy Group Theory in 2005 [6]. Mukherjee [7] developed fuzzy normal subgroups and cosets in 1984. Gupta [5] established the new concept of Theory of T-norms and fuzzy inference methods in 1984. Das [4] introduced the concept of fuzzy groups and level subgroups in 1981. Khare S S[3] introduced the concept of Fuzzy Homomorphism and Algebraic Structures in 1993. Sherwood H [2] established the concept of fuzzy groups in 1979. Rosenfeld A [9] was first suggested by Fuzzy Groups in 1971. Ismai et.al.[8], introduced the concept of Fuzzy Orders Relative to Fuzzy Subgroups and Cyclic group on various fundamental aspects in 2020. In 2016, Abdul Salam[10] described a new notation for a group acting on a fuzzy algebraic structure. Sherwood H [2] established the concept of fuzzy groups in 1979. Dragan Pamucar et.al.[12] introduces the concept of Neutrosophic fuzzy set and its application in decision making in 2020. In 2018, Thiruveni[14], described the notation of Neutrosophic Q-Fuzzy Subgroups. Yager R R [15] Pioneered Fuzzy Sets and Possibility Theory in 1982. Tarnauceanu[13] introduced the concept of classifying fuzzy normal subgroups of finite groups in 2015. Zadeh [16] introduced fuzzy sets for the first time in 1965. Nagarajan [11] introduced the concept of a novel structure and constructed Q-fuzzy groups in 2009. This research introduces the Neutrosophic group operating on fuzzy subsets, extending classical algebraic structures to encompass fuzziness and indeterminacy. And defines the structure and algebraic properties of Neutrosophic group operating fuzzy subgroups and normal subgroups, highlighting how these subsets behave under group operations influenced by uncertainty and indeterminacy. The study also explores homomorphisms, analyzing how fuzzy subgroups behave under structure-preserving mappings, including their images and preimages. Additionally, it presents the direct product of such fuzzy subgroups, proving that key structural properties remain intact across multiple components. Extending this concept to finitely many subgroups, the work enhances the generality and applicability of the model, offering valuable insights into fuzzy algebra and Neutrosophic group theory. 2. Preliminaries Definition 2.1[9]: Assume that, is any group. If, and , then a mapping [0,1] is a fuzzy group. Neutrosophic Sets and Systems, Vol. 97, 2026 577 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Definition 2.2[11]: Let, be a group and be a set. fuzzy set is a mapping in . We define the set for each Ǭ-fuzzy set in and . Definition 2.3[8]: Let, and be any two nonempty sets, with κ ∈ [0,1] and -FSb of a set . The fuzzy set of , also known as the, κ- -FSb of , is defined as . Definition 2.4[12]: A neutrosophic fuzzy set, on the universe of discourse characterized by a truth membership function an indeterminacy function and a falsity membership function is defined as , where , and 3. A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Definition 3.1: Let and be any two non-empty sets, and Let is fuzzy set on for each set on Thus the fuzzy set on as be an group operating fuzzy set in and Then a is and defined by Definition 3.2: Let and be any two non-empty sets, and Let be a in and Then is called of If its following conditions Neutrosophic Sets and Systems, Vol. 97, 2026 578 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup (i) (ii) , Theorem 3.3: Let be of be a in and if and only if , and . Proof: Let be of operating on . Now, (i) , and (ii) and , . And also, , and Neutrosophic Sets and Systems, Vol. 97, 2026 579 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Theorem 3.4: Let be of operating on . Then is of operating on Proof: Let be of operating on Let . Then, Now, Neutrosophic Sets and Systems, Vol. 97, 2026 580 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Hence, is a of operating on Proposition 3.5: Let be of operating on . Then is of operating on Proof: Let be of operating on Let . Then, Now, Neutrosophic Sets and Systems, Vol. 97, 2026 581 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup Hence, is a of operating on Definition 3.6: Let be of is said to be of operating on If , , and , , , . Definition 3.7: Let be of operating on Let with Then is group operating level subset of is defined by Theorem 3.8: If is a of operating on and then is group operating level subset of is a of where Neutrosophic Sets and Systems, Vol. 97, 2026 582 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup where is identity element of operating on and . Proof: Let is a of operating on Since Therefore, Let and . Then, , and and is a of operating on Theorem 3.9: If is a of then is a of if and only if is a of for . Proof: Let Let is of Neutrosophic Sets and Systems, Vol. 97, 2026 583 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup , is a of operating on Theorem 3.10: Let is a of a group operating on Then is a of operating on if and only if is a of operating on Proof: Let be a of operating on and . Hence, is a of operating on 4. A New Constrain of Homomorphism of Neutrosophic Group Operating Fuzzy Subgroup Definition 4.1: Let and be the non-empty sets. Let and be any two groups. The function is said to be group operating homomorphism if (i) is a group operating homomorphism (ii) , and . Neutrosophic Sets and Systems, Vol. 97, 2026 590 Premkumar M, Kiruthika K, J. Saral, D. J. Samatha Naidu, Karuppusamy M, Roselin Suhi R, Usha P and Venkatachalam M, A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgroup 1. Ajmal N. 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Prasanna A, Premkumar M, Ismail Mohideen S and Dhirendra Kumar Shukla, Fuzzy Orders Relative to Fuzzy Subgroups and Cyclic group on various fundamental aspects, Materials Today: Proceedings, 2020, Vol.15, PP: 1-4. 9. Rosenfeld A, Fuzzy Groups, Journal of Mathematical Analysis and Applications, 1971, Vol. 35, PP. 512-517. 10. Solairaju A and Abdul Salam A, A new Construction of a group acting on fuzzy algebraic structure, Advances in fuzzy Mathematics, 2016, Vol. 11, PP. 207-217. 11. Solairaju A and Nagarajan R, A new structure and construction of Q-fuzzy groups, Advances in Fuzzy Mathemtics, 2009, Vol. 4, PP. 23-29. 12. Sujit Das, Bikash Koli Roy, Mohuya B. Kar, Samarjit Kar, Dragan Pamucar, Neutrosophic fuzzy set and its application in decision making, Journal of Ambient Inteklligence and Humanized Computing, 2020, Vol. 11, PP. 5017–5029. 13. Tarnauceanu M, Classifying fuzzy normal subgroups of finite groups, Iranian Journal Fuzzy Systems, 2015, Vol. 12, PP. 107-115. 14. Thiruveni and Solairaju, Neutrosophic Q-Fuzzy Subgroups, International Journal of Mathematics and its Applications, 2018, Vol. 6, PP. 859-866. 15. Yager R R, Fuzzy Sets and Possibility theory, Pergamon, New York, 1982. 16. Zadeh L A, Fuzzy Sets, Information and Control, 165, Vol. 8, PP. 338-353. Received: May 29, 2025. Accepted: Nov 10, 2025