{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:43:45Z","timestamp":1777704225379,"version":"3.51.4"},"reference-count":20,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T00:00:00Z","timestamp":1512086400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p>\n                    One of the problems in fuzzy group theory is concerned with classifying the fuzzy (normal) subgroups of a finite group. In this paper, we classify the fuzzy (normal) subgroups of\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mi>D<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mo>\u00d7<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"double-struck\">\n                              <mml:mi>\u2124<\/mml:mi>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mi>q<\/mml:mi>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    ,\n                    <jats:italic>q<\/jats:italic>\n                    primes) and all finite groups of order\n                    <jats:italic>n<\/jats:italic>\n                    \u00a0\u2264\u00a020, by using the natural equivalence relation. In this case, the corresponding equivalence classes of fuzzy (normal) subgroups of a group\n                    <jats:italic>G<\/jats:italic>\n                    are closely connected to the chains of (normal) subgroups in\n                    <jats:italic>G<\/jats:italic>\n                    . In fact, for determining the number of these classes, we calculate the number of all chains of (normal) subgroups of\n                    <jats:italic>G<\/jats:italic>\n                    that terminate in\n                    <jats:italic>G<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.3233\/jifs-17301","type":"journal-article","created":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T12:36:18Z","timestamp":1512131778000},"page":"3615-3627","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":3,"title":["Classifying fuzzy (normal) subgroups of\u00a0the\u00a0group D2p\u00d7\u2124q and finite groups of\u00a0order\n                    <i>n<\/i>\n                    \u00a0\u2264\u00a020"],"prefix":"10.1177","volume":"33","author":[{"given":"Leili Kamali","family":"Ardekani","sequence":"first","affiliation":[{"name":"Faculty of Engineering, Ardakan University, Ardakan, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bijan","family":"Davvaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yazd University, Yazd, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2017,12]]},"reference":[{"issue":"1","key":"e_1_3_1_2_2","first-page":"209","article-title":"A note on the number of fuzzy subgroups of finite groups","volume":"54","author":"Bentea L.","year":"2008","unstructured":"BenteaL. and T\u0103rn\u0103uceanuM., A note on the number of fuzzy subgroups of finite groups, Annals of the Alexandru Ioan Cuza University - Mathematics54(1) (2008), 209\u2013220.","journal-title":"Annals of the Alexandru Ioan Cuza University - Mathematics"},{"key":"e_1_3_1_3_2","first-page":"175","article-title":"Counting the number of fuzzy subgroups of a special class of non-abelian groups of order p3","volume":"103","author":"Davvaz B.","year":"2012","unstructured":"DavvazB. and Kamali ArdekaniL., Counting the number of fuzzy subgroups of a special class of non-abelian groups of order p3, ARS Combinatoria103 (2012), 175\u2013179.","journal-title":"ARS Combinatoria"},{"issue":"5","key":"e_1_3_1_4_2","first-page":"507","article-title":"Classifying fuzzy sub-groups of dicyclic groups","volume":"20","author":"Davvaz B.","year":"2013","unstructured":"DavvazB. and Kamali ArdekaniL., Classifying fuzzy sub-groups of dicyclic groups, Journal of Multiple-Valued Logic and Soft Computing20(5-6) (2013), 507\u2013525.","journal-title":"Journal of Multiple-Valued Logic and Soft Computing"},{"issue":"5","key":"e_1_3_1_5_2","first-page":"479","article-title":"Counting fuzzy subgroups of non-abelian groups of orders p\n                  3 and 24","volume":"21","author":"Davvaz B.","year":"2013","unstructured":"DavvazB. and Kamali ArdekaniL., Counting fuzzy subgroups of non-abelian groups of orders p3 and 24, Journal of Multiple-Valued Logic and Soft Computing21(5-6) (2013), 479\u2013492.","journal-title":"Journal of Multiple-Valued Logic and Soft Computing"},{"issue":"5","key":"e_1_3_1_6_2","article-title":"Counting fuzzy normal sub-groups of non-abelian finite groups","volume":"28","author":"Davvaz B.","unstructured":"DavvazB. and Ardekani KamaliL., Counting fuzzy normal sub-groups of non-abelian finite groups, Journal of Multiple-Valued Logic and Soft Computing28(5-6. 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