{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:46:10Z","timestamp":1760240770160,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T00:00:00Z","timestamp":1568073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tianjin Sino-German University of Applied Sciences","award":["313\/X18015"],"award-info":[{"award-number":["313\/X18015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Given a positive integer n, a finite group G is called quasi-core-n if      \u27e8 x \u27e9  \/   \u27e8 x \u27e9  G      has order at most n for any element x in G, where      \u27e8 x \u27e9  G     is the normal core of     \u27e8 x \u27e9     in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is     p + m    , then the exponent of its commutator subgroup cannot exceed     p  m + 1     , where p is an odd prime and m is non-negative. If     p = 3    , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.<\/jats:p>","DOI":"10.3390\/sym11091147","type":"journal-article","created":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T10:52:26Z","timestamp":1568112746000},"page":"1147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Finite Quasi-Core-p p-Groups"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6335-3183","authenticated-orcid":false,"given":"Jiao","family":"Wang","sequence":"first","affiliation":[{"name":"Basic Course Department, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiuyun","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1017\/S1446788700037289","article-title":"Groups with all subgroups normal-by-finite","volume":"59","author":"Buckley","year":"1995","journal-title":"J. Aust. Math. Soc."},{"key":"ref_2","unstructured":"Lennox, J.C., Smith, H., and Wiegold, J. (1994, January 23\u201327). Finite p-groups in which subgroups have large cores. Proceedings of the Infinite Groups 1994, International Conference, Ravello, Italy."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1006\/jabr.1996.6811","article-title":"Finite quasi-core-p p-groups","volume":"188","author":"Cutolo","year":"1997","journal-title":"J. Algebra"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1006\/jabr.2000.8599","article-title":"On core-2 2-groups","volume":"237","author":"Cutolo","year":"2001","journal-title":"J. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Huppert, B. (1967). Endliche Gruppen I, Springer.","DOI":"10.1007\/978-3-642-64981-3"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Berkovich, Y. (2008). Groups of Prime Power Order, Volume I, Walter de Gruyter.","DOI":"10.1515\/9783110208221"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3603","DOI":"10.1016\/j.jalgebra.2008.01.045","article-title":"Finite p-groups all of whose non-abelian proper subgroups are generated by two elements","volume":"319","author":"Xu","year":"2008","journal-title":"J. Algebra"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1017\/S0004972700011953","article-title":"A note on regular metabelian groups of prime-power order","volume":"46","author":"Newman","year":"1992","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1142\/S1005386706000058","article-title":"A classification of metacyclic 2-groups","volume":"13","author":"Xu","year":"2006","journal-title":"Algebra Colloq."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:18:23Z","timestamp":1760188703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,10]]},"references-count":9,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091147"],"URL":"https:\/\/doi.org\/10.3390\/sym11091147","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,9,10]]}}}