{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:00:58Z","timestamp":1760241658169,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,6,25]],"date-time":"2018-06-25T00:00:00Z","timestamp":1529884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let G=Zp\u2295Zp2, where p is a prime number. Suppose that d is a divisor of the order of G. In this paper, we find the number of automorphisms of G fixing d elements of G and denote it by \u03b8(G,d). As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group Zpa\u2295Zpb, where a and b are positive integers with a&lt;b. Finally, we compute \u03b8(D2q,d), where D2q is the dihedral group of order 2q, q is an odd prime, and d\u2208{1,q,2q}.<\/jats:p>","DOI":"10.3390\/sym10070238","type":"journal-article","created":{"date-parts":[[2018,6,25]],"date-time":"2018-06-25T11:03:25Z","timestamp":1529924605000},"page":"238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group"],"prefix":"10.3390","volume":"10","author":[{"given":"Umar","family":"Hayat","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9694-3884","authenticated-orcid":false,"given":"Daniel","family":"L\u00f3pez-Aguayo","sequence":"additional","affiliation":[{"name":"Tecnologico de Monterrey, Escuela de Ingenier\u00eda y Ciencias, Hidalgo 42080, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Akhtar","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/1687-1812-2013-55","article-title":"Relationships between fixed points and eigenvectors in the group GL(2,R)","volume":"2013","author":"Demir","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"225204","DOI":"10.1088\/1751-8113\/46\/22\/225204","article-title":"Automorphisms of real Lie Algebras of dimension five or less","volume":"46","author":"Fisher","year":"2013","journal-title":"J. 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Kac Algebras Arising from Composition of Subfactors: General Theory and Classification, Memoirs of the American Mathematical Society.","DOI":"10.1090\/memo\/0750"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:09:58Z","timestamp":1760195398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,25]]},"references-count":11,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["sym10070238"],"URL":"https:\/\/doi.org\/10.3390\/sym10070238","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,6,25]]}}}