{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T05:26:12Z","timestamp":1784784372443,"version":"3.55.0"},"reference-count":56,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Faisal University Saudi Arabia","award":["GRANT 1690"],"award-info":[{"award-number":["GRANT 1690"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group W relative to its normal subgroup S by defining a PFSG of WS. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.<\/jats:p>","DOI":"10.3390\/sym14102084","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2084","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1898-4082","authenticated-orcid":false,"given":"Abdul","family":"Razaq","sequence":"first","affiliation":[{"name":"Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ghaliah","family":"Alhamzi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7579-8972","authenticated-orcid":false,"given":"Asima","family":"Razzaque","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, King Faisal University, Al Ahsa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9099-8422","authenticated-orcid":false,"given":"Harish","family":"Garg","sequence":"additional","affiliation":[{"name":"School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University, Patiala 147004, India"},{"name":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6806","DOI":"10.1109\/TSMC.2021.3099862","article-title":"Multiplicative Consistency Ascertaining, Inconsistency Repairing, and Weights Derivation of Hesitant Multiplicative Preference Relations","volume":"52","author":"Xu","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"178","DOI":"10.37394\/23206.2021.20.19","article-title":"Generating Fuzzy Sets and Fuzzy Relations Based on Information","volume":"20","author":"Noaman","year":"2021","journal-title":"WSEAS Trans. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"624","DOI":"10.37394\/23206.2022.21.72","article-title":"Topological Spaces on Fuzzy Structures","volume":"21","author":"Voskoglou","year":"2022","journal-title":"WSEAS Trans. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"31","DOI":"10.37394\/232021.2022.2.6","article-title":"Usage of the Fuzzy Laplace Transform Method for Solving One-Dimensional Fuzzy Integral Equations","volume":"2","author":"Sharif","year":"2022","journal-title":"EQUATIONS"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3498","DOI":"10.1109\/TSMC.2019.2931536","article-title":"Algorithms to Detect and Rectify Multiplicative and Ordinal Inconsistencies of Fuzzy Preference Relations","volume":"51","author":"Xu","year":"2019","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanasov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3980","DOI":"10.1109\/TSMC.2019.2958635","article-title":"A Distance Measure for Intuitionistic Fuzzy Sets and Its Application to Pattern Classification Problems","volume":"51","author":"Xiao","year":"2019","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Shi, Y., Chu, J., Ji, C., Li, J., and Ning, S. (2022). A Fuzzy-Based Mobile Edge Architecture for Latency-Sensitive and Heavy-Task Applications. Symmetry, 14.","DOI":"10.3390\/sym14081667"},{"key":"ref_10","first-page":"42","article-title":"A new study based on fuzzy bi-\u0393-ideals in ordered-\u0393-semigroups","volume":"1","author":"Muhiuddin","year":"2022","journal-title":"J. Comput. Cogn. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Song, Y., Fu, Q., Wang, Y.-F., and Wang, X. (2019). Divergence-based cross entropy and uncertainty measures of Atanassov\u2019s intuitionistic fuzzy sets with their application in decision making. Appl. Soft Comput., 84.","DOI":"10.1016\/j.asoc.2019.105703"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4959","DOI":"10.1007\/s00500-018-3202-1","article-title":"An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making","volume":"22","author":"Garg","year":"2018","journal-title":"Soft Comput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2013, January 24\u201328). Pythagorean fuzzy subsets. Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hussain, A., Ullah, K., Alshahrani, M.N., Yang, M.-S., and Pamucar, D. (2022). Novel Aczel\u2013Alsina Operators for Pythagorean Fuzzy Sets with Application in Multi-Attribute Decision Making. Symmetry, 14.","DOI":"10.3390\/sym14050940"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhou, Q., Mo, H., and Deng, Y. (2020). A New Divergence Measure of Pythagorean Fuzzy Sets Based on Belief Function and Its Application in Medical Diagnosis. Mathematics, 8.","DOI":"10.3390\/math8010142"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1002\/int.21979","article-title":"Linguistic Pythagorean fuzzy sets and its applications in multi-attribute decision-making process","volume":"33","author":"Garg","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Silva, A.J.V., Dominguez, L.A.P., G\u00f3mez, E.M., Alvarado-Iniesta, A., and Olgu\u00edn, I.J.C.P. (2019). Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection. Symmetry, 11.","DOI":"10.3390\/sym11030336"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.3233\/JIFS-182540","article-title":"A new fuzzy entropy on Pythagorean fuzzy sets","volume":"37","author":"Thao","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_19","first-page":"108","article-title":"Pythagorean neutrosophic fuzzy graphs","volume":"11","author":"Ajay","year":"2020","journal-title":"Int. J. Neutrosophic Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Garg, H. (2021). Pythagorean Fuzzy Sets: Theory & Applications, Springer.","DOI":"10.1007\/978-981-16-1989-2"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Munde, A. (2022). Pythagorean Fuzzy Information Measure with Application to Multicriteria Decision Making. Congress on Intelligent Systems, Springer.","DOI":"10.1007\/978-981-16-9416-5_6"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Xu, Y., Zhu, S., Liu, X., Huang, J., and Herrera-Viedma, E. (2022). Additive consistency exploration of linguistic preference relations with self-confidence. Artif. Intell. Rev., 1\u201329.","DOI":"10.1007\/s10462-022-10172-x"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"108438","DOI":"10.1016\/j.cie.2022.108438","article-title":"An improved HK model-driven consensus reaching for group decision making under interval-valued fuzzy preference relations with self-confidence","volume":"171","author":"Li","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.ins.2019.07.050","article-title":"Social network group decision making: Managing self-confidence-based consensus model with the dynamic importance degree of experts and trust-based feedback mechanism","volume":"505","author":"Liu","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_25","first-page":"12","article-title":"What modern algebra is about","volume":"3","author":"Hall","year":"1940","journal-title":"Eureka"},{"key":"ref_26","first-page":"1","article-title":"Quantenmechanik und Gruppentheorie","volume":"46","author":"Weyl","year":"1927","journal-title":"Eur. Phys. J. A"},{"key":"ref_27","unstructured":"Cotton, F.A. (1991). The Intuitionistic Fuzzy Normal Subgroup and Its Some Equivalent Propositions, John Wiley & Sons."},{"key":"ref_28","unstructured":"Inui, T., Tanabe, Y., and Onodera, Y. (2012). Group Theory and Its Applications in Physics, Springer."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ivanov, A.A., and Shpectorov, S.V. (1994). Applications of group amalgams to algebraic graph theory. Investigations in Algebraic Theory of Combinatorial Objects, Springer.","DOI":"10.1007\/978-94-017-1972-8_14"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3905425","DOI":"10.1155\/2021\/3905425","article-title":"A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group","volume":"2021","author":"Alolaiyan","year":"2021","journal-title":"J. Math."},{"key":"ref_31","first-page":"21","article-title":"Applications of group theory in cryptography","volume":"58","author":"Kahrobaei","year":"2010","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4163","DOI":"10.1007\/s11042-021-11635-z","article-title":"A group theoretic construction of highly nonlinear sub-stitution box and its applications in image encryption","volume":"81","author":"Razaq","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1007\/s11277-021-08981-4","article-title":"A Group Theoretic Construction of Large Number of AES-Like Substitution-Boxes","volume":"122","author":"Razaq","year":"2021","journal-title":"Wirel. Pers. Commun."},{"key":"ref_34","unstructured":"Olver, P.J. (2000). Applications of Lie Groups to Differential Equations, Springer."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Hamermesh, M. (2005). Group theory. Mathematical Tools for Physicists, Wiley.","DOI":"10.1002\/3527607773.ch7"},{"key":"ref_36","unstructured":"Kettle, S.F. (2008). Symmetry and Structure: Readable Group Theory for Chemists, John Wiley & Sons."},{"key":"ref_37","unstructured":"McWeeny, R. (2002). Symmetry: An Introduction to Group Theory and Its Applications, Courier Corporation."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Powell, R.C. (2010). Symmetry, Group Theory, and the Physical Properties of Crystals, Springer.","DOI":"10.1007\/978-1-4419-7598-0"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/0022-247X(71)90199-5","article-title":"Fuzzy groups","volume":"35","author":"Rosenfeld","year":"1971","journal-title":"J. Math. Anal. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/0022-247X(81)90164-5","article-title":"Fuzzy groups and level subgroups","volume":"84","author":"Das","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/0165-0114(82)90003-3","article-title":"Fuzzy invariant subgroups and fuzzy ideals","volume":"8","author":"Liu","year":"1982","journal-title":"Fuzzy Sets Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0020-0255(84)90050-1","article-title":"Fuzzy normal subgroups and fuzzy cosets","volume":"34","author":"Mukherjee","year":"1984","journal-title":"Inf. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0020-0255(86)90039-3","article-title":"Fuzzy groups: Some group-theoretic analogs","volume":"39","author":"Mukherjee","year":"1986","journal-title":"Inf. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/0165-0114(85)90027-2","article-title":"Semidirect products OF fuzzy subgroups","volume":"16","author":"Wetherilt","year":"1985","journal-title":"Fuzzy Sets Syst."},{"key":"ref_45","first-page":"53","article-title":"Normal fuzzy subgroups","volume":"20","author":"Mashour","year":"1990","journal-title":"Inf. Sci."},{"key":"ref_46","first-page":"1","article-title":"Fuzzy normal subgroups in fuzzy subgroups","volume":"29","author":"Malik","year":"1992","journal-title":"J. Korean Math. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0165-0114(90)90032-2","article-title":"Level subgroups and union of fuzzy subgroups","volume":"37","author":"Dixit","year":"1990","journal-title":"Fuzzy Sets Syst."},{"key":"ref_48","first-page":"39","article-title":"Intuitionistic fuzzy subgroups","volume":"10","author":"Biswas","year":"1989","journal-title":"Math. Forum"},{"key":"ref_49","first-page":"17","article-title":"Intuitionistic fuzzy subgroups and cosets","volume":"26","author":"Hur","year":"2004","journal-title":"Honam Math. J."},{"key":"ref_50","first-page":"523","article-title":"Sharma on the direct product of intuitionistic fuzzy subgroups","volume":"7","author":"Sharma","year":"2012","journal-title":"Int. Math. Forum"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Altassan, A., Mateen, M., and Pamucar, D. (2021). On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product. Symmetry, 13.","DOI":"10.3390\/sym13060998"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Alharbi, A.A., and Alghazzawi, D. (2022). Some Characterizations of Certain Complex Fuzzy Subgroups. Symmetry, 14.","DOI":"10.3390\/sym14091812"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"158419","DOI":"10.1109\/ACCESS.2019.2950167","article-title":"t-Intuitionistic Fuzzification of Lagrange\u2019s Theorem of t-Intuitionistic Fuzzy Subgroup","volume":"7","author":"Alolaiyan","year":"2019","journal-title":"IEEE Access"},{"key":"ref_54","first-page":"40","article-title":"The intuitionistic fuzzy normal subgroup and its some equivalent propositions","volume":"82","author":"Li","year":"2000","journal-title":"Busefal"},{"key":"ref_55","first-page":"40","article-title":"Intuitionistic fuzzy subgroups with respect to norms (T, S)","volume":"3","author":"Rasuli","year":"2020","journal-title":"Eng. Appl. Sci. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"962","DOI":"10.3934\/math.2021058","article-title":"On the characterization of Pythagorean fuzzy subgroups","volume":"6","author":"Bhunia","year":"2021","journal-title":"AIMS Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2084\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:47:34Z","timestamp":1760143654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2084"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,7]]},"references-count":56,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102084"],"URL":"https:\/\/doi.org\/10.3390\/sym14102084","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,7]]}}}