{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:04Z","timestamp":1760149744720,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T00:00:00Z","timestamp":1693785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University","award":["RSPD2023R650"],"award-info":[{"award-number":["RSPD2023R650"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this research article, the fundamental properties of complex anti-fuzzy subgroups as well as the influence of group homomorphisms on their characteristics are investigated. Both the necessary and sufficient conditions for a complex anti-fuzzy subgroup are defined; additionally, the image, inverse image and some vital primary features of complex anti-fuzzy subgroups are examined. Moreover, the homomorphic and isomorphic relations of complex anti-fuzzy subgroups under group homomorphism are discussed, and numerical examples for various scenarios to describe complex anti-fuzzy symmetric groups are provided. Finally, it is proven that every homomorphic image of a complex anti-fuzzy cyclic group is cyclic, but the converse may not be true.<\/jats:p>","DOI":"10.3390\/sym15091693","type":"journal-article","created":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T10:14:11Z","timestamp":1693822451000},"page":"1693","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Novel Concept of Complex Anti-Fuzzy Isomorphism over Groups"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6220-4373","authenticated-orcid":false,"given":"Bander","family":"Almutairi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arshad","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National College of Business Administration and Economics, Lahore 54660, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6178-8538","authenticated-orcid":false,"given":"Qin","family":"Xin","sequence":"additional","affiliation":[{"name":"Faculty of Science and Technology, University of the Faroe Islands, FO 100 Torshavn, Faroe Islands, Denmark"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1490-8576","authenticated-orcid":false,"given":"Adnan","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National College of Business Administration and Economics, Lahore 54660, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3823","DOI":"10.1021\/acs.chemmater.9b05324","article-title":"Search for Ferroelectric Binary Oxides: Chemical and Structural Space Exploration Guided by Group Theory and Computations","volume":"32","author":"Batra","year":"2020","journal-title":"Chem. 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