{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T11:05:06Z","timestamp":1774868706144,"version":"3.50.1"},"reference-count":16,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T00:00:00Z","timestamp":1756684800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    We formalize the semidirect product of groups in Mizar, following \u00a710 of Aschbacher\u2019s\n                    <jats:italic>Finite Group Theory<\/jats:italic>\n                    [2]. We also prove the universal property for semidirect products as found in Bourbaki [5, III \u00a72.10] Proposition 27. In an appendix, we define the dihedral group of the regular\n                    <jats:italic>n<\/jats:italic>\n                    -gon and the infinite dihedral group.\n                  <\/jats:p>","DOI":"10.2478\/forma-2025-0006","type":"journal-article","created":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T12:05:49Z","timestamp":1765973149000},"page":"65-83","source":"Crossref","is-referenced-by-count":0,"title":["Semidirect Products of Groups"],"prefix":"10.2478","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3292-4035","authenticated-orcid":false,"given":"Alexander M.","family":"Nelson","sequence":"first","affiliation":[{"name":"Los Angeles , California United States of America"}]}],"member":"374","published-online":{"date-parts":[[2025,12,17]]},"reference":[{"key":"2026033010152360429_j_forma-2025-0006_ref_001","doi-asserted-by":"crossref","unstructured":"Goulnara Arzhantseva and \u015awiatos\u0142aw R. Gal. On approximation properties of semidirect products of groups. Annales Math\u00e9matiques Blaise Pascal, 27(1):1\u201324, 2020. doi:10.5802\/ambp.386.","DOI":"10.5802\/ambp.386"},{"key":"2026033010152360429_j_forma-2025-0006_ref_002","doi-asserted-by":"crossref","unstructured":"Michael Aschbacher. Finite Group Theory, volume 10. Cambridge University Press, 2000.","DOI":"10.1017\/CBO9781139175319"},{"key":"2026033010152360429_j_forma-2025-0006_ref_003","doi-asserted-by":"crossref","unstructured":"Clia Borlido and Mai Gehrke. Substitution principle and semidirect products. Mathematical Structures in Computer Science, 33(6):486\u2013535, 2023. doi:10.1017\/S0960129523000294.","DOI":"10.1017\/S0960129523000294"},{"key":"2026033010152360429_j_forma-2025-0006_ref_004","unstructured":"Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989."},{"key":"2026033010152360429_j_forma-2025-0006_ref_005","unstructured":"Nicolas Bourbaki. General Topology: Chapters 1\u20134. Springer Science and Business Media, 2013."},{"key":"2026033010152360429_j_forma-2025-0006_ref_006","unstructured":"Peteris Daugulis. Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups. Algebra and Discrete Mathematics, 23(2):204\u2013215, 2017."},{"key":"2026033010152360429_j_forma-2025-0006_ref_007","unstructured":"David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley and Sons, Third edition, 2004."},{"key":"2026033010152360429_j_forma-2025-0006_ref_008","doi-asserted-by":"crossref","unstructured":"Wolfgang Gasch\u00fctz. Zu einem von B. H. und H. Neumann gestellten Problem. Mathematische Nachrichten, 14(4\u20136):249\u2013252, 1955. doi:10.1002\/mana.19550140406.","DOI":"10.1002\/mana.19550140406"},{"key":"2026033010152360429_j_forma-2025-0006_ref_009","doi-asserted-by":"crossref","unstructured":"Craig R. Guilbault, Brendan Burns Healy, and Brian Pietsch. Group boundaries for semi-direct products with \u2124. Groups, Geometry, and Dynamics, 2024. doi:10.4171\/GGD\/750.","DOI":"10.4171\/ggd\/750"},{"key":"2026033010152360429_j_forma-2025-0006_ref_010","unstructured":"Scott Harper and Peiran Wu. Classifying the groups of order pq in Lean. arXiv preprint arXiv:2501.09769, 2025."},{"key":"2026033010152360429_j_forma-2025-0006_ref_011","doi-asserted-by":"crossref","unstructured":"I. Martin Isaacs. Finite Group Theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, 2008.","DOI":"10.1090\/gsm\/092"},{"key":"2026033010152360429_j_forma-2025-0006_ref_012","unstructured":"Vipul Kakkar and Ratan Lal. Automorphisms of semidirect products fixing the non-normal subgroup. Jordan Journal of Mathematics and Statistics, 17(2):241\u2013248, 2024. doi:10.47013\/17.2.5."},{"key":"2026033010152360429_j_forma-2025-0006_ref_013","unstructured":"Artur Korni\u0142owicz. On the group of inner automorphisms. Formalized Mathematics, 5 (1):43\u201345, 1996."},{"key":"2026033010152360429_j_forma-2025-0006_ref_014","unstructured":"Artur Korni\u0142owicz. The product of the families of the groups. Formalized Mathematics, 7(1):127\u2013134, 1998."},{"key":"2026033010152360429_j_forma-2025-0006_ref_015","doi-asserted-by":"crossref","unstructured":"Alexander M. Nelson. Internal direct products and the universal property of direct product groups. Formalized Mathematics, 31(1):101\u2013120, 2023. doi:10.2478\/forma-2023-0010.","DOI":"10.2478\/forma-2023-0010"},{"key":"2026033010152360429_j_forma-2025-0006_ref_016","doi-asserted-by":"crossref","unstructured":"Colin Rothgang, Artur Korni\u0142owicz, and Florian Rabe. A new export of the Mizar Mathematical Library. In Fairouz Kamareddine and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics, pages 205\u2013210, Cham, 2021. Springer International Publishing. doi:10.1007\/978-3-030-81097-9 17.","DOI":"10.1007\/978-3-030-81097-9_17"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.sciendo.com\/pdf\/10.2478\/forma-2025-0006","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T10:16:04Z","timestamp":1774865764000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.2478\/forma-2025-0006"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,1]]},"references-count":16,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12,17]]},"published-print":{"date-parts":[[2025,9,1]]}},"alternative-id":["10.2478\/forma-2025-0006"],"URL":"https:\/\/doi.org\/10.2478\/forma-2025-0006","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,1]]}}}