{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T19:06:00Z","timestamp":1754161560607,"version":"3.41.2"},"reference-count":8,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:p> Given a group G and positive integers [Formula: see text], we let [Formula: see text] denote the set of all elements with at most n conjugates in [Formula: see text] We say that G satisfies the [Formula: see text]-covering condition for commutators if there is a subset [Formula: see text] such that [Formula: see text] and all commutators of G are contained in [Formula: see text]. The importance of groups satisfying this condition was revealed in the recent study of probabilistically nilpotent finite groups of class two. The main result obtained in this paper is the following theorem. Let G be a group satisfying the [Formula: see text]-covering condition for commutators. Then [Formula: see text] contains a characteristic subgroup W such that [Formula: see text] and [Formula: see text] are both [Formula: see text]-bounded. This extends several earlier results of similar flavor. <\/jats:p>","DOI":"10.1142\/s0218196725500250","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T10:42:43Z","timestamp":1748601763000},"page":"759-770","source":"Crossref","is-referenced-by-count":0,"title":["Groups with a covering condition on commutators"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3091-9857","authenticated-orcid":false,"given":"Eloisa","family":"Detomi","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica \u201cTullio Levi-Civita\u201d, Universit\u00e0 di Padova, Via Trieste 63, 35121 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9066-8818","authenticated-orcid":false,"given":"Marta","family":"Morigi","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bologna, Piazza di Porta San Donato 5, 40126 Bologna Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4976-5675","authenticated-orcid":false,"given":"Pavel","family":"Shumyatsky","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2025,7,1]]},"reference":[{"key":"S0218196725500250BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2024.06.031"},{"key":"S0218196725500250BIB002","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/hay014"},{"key":"S0218196725500250BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-023-02567-0"},{"key":"S0218196725500250BIB004","first-page":"298","volume":"226","author":"Guralnick R. M.","year":"2011","journal-title":"J. Algebra"},{"key":"S0218196725500250BIB005","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-29.2.236"},{"volume-title":"The Algebraic Theory of Group Rings","year":"1977","author":"Passman D.","key":"S0218196725500250BIB006"},{"key":"S0218196725500250BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8594-1"},{"key":"S0218196725500250BIB008","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1957.0007"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196725500250","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T02:57:14Z","timestamp":1753757834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196725500250"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,1]]},"references-count":8,"journal-issue":{"issue":"05","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["10.1142\/S0218196725500250"],"URL":"https:\/\/doi.org\/10.1142\/s0218196725500250","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"type":"print","value":"0218-1967"},{"type":"electronic","value":"1793-6500"}],"subject":[],"published":{"date-parts":[[2025,7,1]]}}}