{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:15:49Z","timestamp":1760238949870,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,17]],"date-time":"2020-09-17T00:00:00Z","timestamp":1600300800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771271"],"award-info":[{"award-number":["11771271"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.<\/jats:p>","DOI":"10.3390\/sym12091537","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T10:22:23Z","timestamp":1600424543000},"page":"1537","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Number of Subgroup Chains of Finite Nilpotent Groups"],"prefix":"10.3390","volume":"12","author":[{"given":"Lingling","family":"Han","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2288-2983","authenticated-orcid":false,"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":[[2020,9,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Aigner, M. 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Comb."},{"key":"ref_6","first-page":"65","article-title":"The number of chains of subgroups of a finite elementary abelian p-group","volume":"77","year":"2015","journal-title":"Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2623","DOI":"10.1016\/j.disc.2007.05.017","article-title":"A note on lattice chains and Delannoy numbers","volume":"308","author":"Caughman","year":"2008","journal-title":"Discrete Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.ejc.2011.10.004","article-title":"The number of chains of subgroups of a finite cyclic group","volume":"33","author":"Oh","year":"2012","journal-title":"Eur. J. Comb."},{"key":"ref_9","first-page":"163","article-title":"Counting distinct fuzzy subgroups of some rank-3 abelian groups","volume":"14","author":"Appiah","year":"2017","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0165-0114(02)00140-9","article-title":"On an equivalence of fuzzy subgroups, II","volume":"136","author":"Murali","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/S0165-0114(03)00224-0","article-title":"Counting the number of fuzzy subgroups of an abelian group of order pnqm","volume":"144","author":"Murali","year":"2004","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","first-page":"125","article-title":"An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two","volume":"10","author":"Oh","year":"2013","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_13","first-page":"149","article-title":"Fuzzy subgroups of rank two abelian p-group","volume":"7","author":"Ngcibi","year":"2010","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1084","DOI":"10.1016\/j.fss.2007.11.014","article-title":"On the number of fuzzy subgroups of finite abelian groups","volume":"159","author":"Bentea","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","first-page":"87","article-title":"Fuzzy subgroups and chains of subgroups","volume":"10","author":"Volf","year":"2004","journal-title":"FSAI"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1537\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:10:57Z","timestamp":1760177457000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1537"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,17]]},"references-count":15,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091537"],"URL":"https:\/\/doi.org\/10.3390\/sym12091537","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,9,17]]}}}