{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T16:55:29Z","timestamp":1755795329052},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T00:00:00Z","timestamp":1656633600000},"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":[[2022,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We formalize in Mizar [1], [2] the notion of characteristic subgroups using the definition found in Dummit and Foote [3], as subgroups invariant under automorphisms from its parent group. Along the way, we formalize notions of Automorphism and results concerning centralizers. Much of what we formalize may be found sprinkled throughout the literature, in particular Gorenstein [4] and Isaacs [5]. We show all our favorite subgroups turn out to be characteristic: the center, the derived subgroup, the commutator subgroup generated by characteristic subgroups, and the intersection of all subgroups satisfying a generic group property.<\/jats:p>","DOI":"10.2478\/forma-2022-0007","type":"journal-article","created":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T10:16:49Z","timestamp":1671877009000},"page":"79-91","source":"Crossref","is-referenced-by-count":1,"title":["Characteristic Subgroups"],"prefix":"10.2478","volume":"30","author":[{"given":"Alexander M.","family":"Nelson","sequence":"first","affiliation":[{"name":"Los Angeles, California , United States of America"}]}],"member":"374","published-online":{"date-parts":[[2022,12,24]]},"reference":[{"key":"2023042002583877686_j_forma-2022-0007_ref_001","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2023042002583877686_j_forma-2022-0007_ref_002","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2023042002583877686_j_forma-2022-0007_ref_003","unstructured":"[3] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley and Sons, Third edition, 2004."},{"key":"2023042002583877686_j_forma-2022-0007_ref_004","unstructured":"[4] Daniel Gorenstein. Finite Groups. Chelsea Publishing Company, Second edition, 1980."},{"key":"2023042002583877686_j_forma-2022-0007_ref_005","doi-asserted-by":"crossref","unstructured":"[5] I. Martin Isaacs. Finite Group Theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, 2008.10.1090\/gsm\/092","DOI":"10.1090\/gsm\/092"},{"key":"2023042002583877686_j_forma-2022-0007_ref_006","doi-asserted-by":"crossref","unstructured":"[6] Marco Riccardi. The Jordan-H\u00f6lder theorem. Formalized Mathematics, 15(2):35\u201351, 2007. doi:10.2478\/v10037-007-0005-8.","DOI":"10.2478\/v10037-007-0005-8"},{"key":"2023042002583877686_j_forma-2022-0007_ref_007","unstructured":"[7] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41\u201347, 1991."},{"key":"2023042002583877686_j_forma-2022-0007_ref_008","unstructured":"[8] Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145\u2013147, 1996."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.sciendo.com\/pdf\/10.2478\/forma-2022-0007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T16:23:10Z","timestamp":1682007790000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.2478\/forma-2022-0007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,1]]},"references-count":8,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,12,24]]},"published-print":{"date-parts":[[2022,7,1]]}},"alternative-id":["10.2478\/forma-2022-0007"],"URL":"https:\/\/doi.org\/10.2478\/forma-2022-0007","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,1]]}}}