{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:24:22Z","timestamp":1760239462321,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,26]],"date-time":"2020-11-26T00:00:00Z","timestamp":1606348800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971493"],"award-info":[{"award-number":["11971493"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this research paper, we deal with the problem of determining the function \u03c7:G\u2192R, which is the solution to the maximum functional equation (MFE) max{\u03c7(xy),\u03c7(xy\u22121)}=\u03c7(x)\u03c7(y), when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation max{\u03c7(xy),\u03c7(xy\u22121)}=\u03c7(x)+\u03c7(y) and its solutions for the function \u03c7:G\u2192R, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.<\/jats:p>","DOI":"10.3390\/sym12121949","type":"journal-article","created":{"date-parts":[[2020,11,26]],"date-time":"2020-11-26T22:00:33Z","timestamp":1606428033000},"page":"1949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Stability of Maximum Functional Equation and Some Properties of Groups"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4978-9287","authenticated-orcid":false,"given":"Muhammad","family":"Sarfraz","sequence":"first","affiliation":[{"name":"School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6049-5282","authenticated-orcid":false,"given":"Qi","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4322-308X","authenticated-orcid":false,"given":"Yongjin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,26]]},"reference":[{"key":"ref_1","first-page":"99191049","article-title":"Charakterisierung des Betrages reellwertiger additiver Funktionen auf Gruppen","volume":"4","author":"Volkmann","year":"2017","journal-title":"KITopen"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s00013-017-1061-0","article-title":"On the functional equation f(x) + f(y) = max{f(xy),f(xy\u22121)} on groups","volume":"109","author":"Toborg","year":"2017","journal-title":"Arch. 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Math."},{"key":"ref_6","first-page":"11","article-title":"On functional equations in connection with the absolute value of additive functions","volume":"32","author":"Jarczyk","year":"2010","journal-title":"Ser. Math. Catoviciensis Debr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00010-013-0206-x","article-title":"On Tabor groupoids and stability of some functional equations","volume":"87","author":"Badora","year":"2014","journal-title":"Aequ. Math."},{"key":"ref_8","first-page":"4","article-title":"Superstability of some functional equation","volume":"31","author":"Przebieracz","year":"2010","journal-title":"Ser. Math. Catoviciensis Debr."},{"key":"ref_9","first-page":"1","article-title":"The stability of functional equation min{f(x + y),f(x \u2212 y)} = |f(x) \u2212 f(y)|","volume":"1","author":"Przebieracz","year":"2011","journal-title":"J. Inequal. Appl."},{"key":"ref_10","first-page":"69","article-title":"The functional equation f(xy) + f(xy\u22121) = 2f(x)f(y) for groups","volume":"19","author":"Kannappan","year":"1968","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1090\/S0002-9939-1985-0770551-0","article-title":"A characterization of free abelian groups","volume":"93","year":"1985","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1238","DOI":"10.4153\/CJM-1969-136-1","article-title":"A classification of n-abelian groups","volume":"21","author":"Alperin","year":"1969","journal-title":"Can. J. Math."},{"key":"ref_13","first-page":"15","article-title":"Factorization of n-soluble and n-nilpotent groups","volume":"4","author":"Baer","year":"1953","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1112\/jlms\/s1-24.4.247","article-title":"Embedding theorems for groups","volume":"1","author":"Higman","year":"1949","journal-title":"J. Lond. Math. Soc."},{"key":"ref_15","unstructured":"Magnus, W., Karrass, A., and Solitar, D. (2004). Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Courier Corporation."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1949\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:37:28Z","timestamp":1760179048000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1949"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,26]]},"references-count":15,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12121949"],"URL":"https:\/\/doi.org\/10.3390\/sym12121949","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,11,26]]}}}