{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T09:10:02Z","timestamp":1756717802196,"version":"3.44.0"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"5","license":[{"start":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T00:00:00Z","timestamp":1743120000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00144\/2020","UIDP\/00144\/2020"],"award-info":[{"award-number":["UIDB\/00144\/2020","UIDP\/00144\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T017619\/1"],"award-info":[{"award-number":["EP\/T017619\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We determine the closure of a cyclic subgroup \ud835\udc3b of a free group for the pro-\ud835\udc15 topology when \ud835\udc15 is an extension-closed pseudovariety of finite groups.\nWe show that \ud835\udc3b is always closed for the pro-nilpotent topology and compute its closure for the pro-<jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"bold\">G<\/m:mi>\n                              <m:mi>p<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2023-0081_ineq_0001.png\"\/>\n                        <jats:tex-math>\\mathbf{G}_{p}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and pro-<jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"bold\">V<\/m:mi>\n                              <m:mi>p<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2023-0081_ineq_0002.png\"\/>\n                        <jats:tex-math>\\mathbf{V}_{p}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> topologies, where <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"bold\">G<\/m:mi>\n                              <m:mi>p<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2023-0081_ineq_0001.png\"\/>\n                        <jats:tex-math>\\mathbf{G}_{p}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is the pseudovariety of finite \ud835\udc5d-groups and <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi mathvariant=\"bold\">V<\/m:mi>\n                                 <m:mi>p<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>\u2283<\/m:mo>\n                              <m:msub>\n                                 <m:mi mathvariant=\"bold\">G<\/m:mi>\n                                 <m:mi>p<\/m:mi>\n                              <\/m:msub>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2023-0081_ineq_0004.png\"\/>\n                        <jats:tex-math>\\mathbf{V}_{p}\\supset\\mathbf{G}_{p}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> denote the pseudovarieties involved in the join decomposition of the pseudovariety of finite supersolvable groups.\nMore generally, given any nonempty set \ud835\udc43 of primes, we consider the stated problem for the pseudovariety <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"bold\">G<\/m:mi>\n                              <m:mi>P<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2023-0081_ineq_0005.png\"\/>\n                        <jats:tex-math>\\mathbf{G}_{P}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of all finite groups having order a product of primes in \ud835\udc43.<\/jats:p>","DOI":"10.1515\/jgth-2023-0081","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T05:28:32Z","timestamp":1743139712000},"page":"1115-1130","source":"Crossref","is-referenced-by-count":0,"title":["On the closure of cyclic subgroups  of a free group in some profinite topologies"],"prefix":"10.1515","volume":"28","author":[{"given":"Claude","family":"Marion","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica , Faculdade de Ci\u00eancias , Universidade do Porto , R. Campo Alegre 687, 4169-007 Porto , Portugal"}]},{"given":"Pedro V.","family":"Silva","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica , Faculdade de Ci\u00eancias , Universidade do Porto , R. Campo Alegre 687, 4169-007 Porto , Portugal"}]},{"given":"Gareth","family":"Tracey","sequence":"additional","affiliation":[{"name":"Mathematics Institute , University of Warwick , Coventry CV4 7AL , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2025,3,28]]},"reference":[{"key":"2025090108553492485_j_jgth-2023-0081_ref_001","doi-asserted-by":"crossref","unstructured":"J. Almeida, S.\u2009W. Margolis and M.\u2009V. Volkov,\nThe pseudovariety of semigroups of triangular matrices over a finite field,\nTheor. Inform. Appl. 39 (2005), no. 1, 31\u201348.","DOI":"10.1051\/ita:2005002"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_002","doi-asserted-by":"crossref","unstructured":"K. Auinger and B. Steinberg,\nVarieties of finite supersolvable groups with the M. Hall property,\nMath. Ann. 335 (2006), no. 4, 853\u2013877.","DOI":"10.1007\/s00208-006-0767-2"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_003","doi-asserted-by":"crossref","unstructured":"F. Berlai and M. Ferov,\nSeparating cyclic subgroups in graph products of groups,\nJ. Algebra 531 (2019), 19\u201356.","DOI":"10.1016\/j.jalgebra.2019.05.001"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_004","doi-asserted-by":"crossref","unstructured":"M. Hall, Jr.,\nCoset representations in free groups,\nTrans. Amer. Math. Soc. 67 (1949), 421\u2013432.","DOI":"10.1090\/S0002-9947-1949-0032642-4"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_005","doi-asserted-by":"crossref","unstructured":"M. Hall, Jr.,\nA topology for free groups and related groups,\nAnn. of Math. (2) 52 (1950), 127\u2013139.","DOI":"10.2307\/1969513"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_006","doi-asserted-by":"crossref","unstructured":"J. Huang, M. Pawliuk, M. Sabok and D.\u2009T. Wise,\nThe Hrushovski property for hypertournaments and profinite topologies,\nJ. Lond. Math. Soc. (2) 100 (2019), no. 3, 757\u2013774.","DOI":"10.1112\/jlms.12244"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_007","doi-asserted-by":"crossref","unstructured":"K. Iwasawa,\nEinige S\u00e4tze \u00fcber freie Gruppen,\nProc. Acad. Tokyo 19 (1943), 272\u2013274.","DOI":"10.3792\/pia\/1195573488"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_008","doi-asserted-by":"crossref","unstructured":"S. Margolis, M. Sapir and P. Weil,\nClosed subgroups in pro-\ud835\udc15 topologies and the extension problem for inverse automata,\nInternat. J. Algebra Comput. 11 (2001), no. 4, 405\u2013445.","DOI":"10.1142\/S0218196701000498"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_009","unstructured":"C. Marion, P.\u2009V. Silva and G. Tracey,\nOn the pseudovariety of groups \n                  \n                     \n                        \n                           U\n                           =\n                           \n                              \n                                 \u22c1\n                                 \n                                    p\n                                    \u2208\n                                    P\n                                 \n                              \n                              \n                                 \n                                    \n                                       Ab\n                                       \u2062\n                                       \n                                          (\n                                          p\n                                          )\n                                       \n                                    \n                                    \u2217\n                                    Ab\n                                 \n                                 \u2062\n                                 \n                                    (\n                                    \n                                       p\n                                       \u2212\n                                       1\n                                    \n                                    )\n                                 \n                              \n                           \n                        \n                     \n                     \n                     \\mathbf{U}=\\bigvee_{p\\in\\mathbb{P}}\\mathbf{Ab}(p)\\ast\\mathbf{Ab}(p-1)\n                  \n               , preprint (2023), https:\/\/arxiv.org\/abs\/2304.10522."},{"key":"2025090108553492485_j_jgth-2023-0081_ref_010","unstructured":"C. Marion, P.\u2009V. Silva and G. Tracey,\nThe pro-\ud835\udc58-solvable topology on a free group, preprint (2023), https:\/\/arxiv.org\/abs\/2304.10235."},{"key":"2025090108553492485_j_jgth-2023-0081_ref_011","doi-asserted-by":"crossref","unstructured":"C. Marion, P.\u2009V. Silva and G. Tracey,\nThe pro-supersolvable topology on a free group: Deciding denseness,\nJ. Algebra 646 (2024), 183\u2013204.","DOI":"10.1016\/j.jalgebra.2024.02.002"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_012","doi-asserted-by":"crossref","unstructured":"H. Neumann,\nVarieties of Groups,\nSpringer, Berlin, 1967.","DOI":"10.1007\/978-3-642-88599-0"},{"key":"2025090108553492485_j_jgth-2023-0081_ref_013","doi-asserted-by":"crossref","unstructured":"L. Ribes and P.\u2009A. Zalesskii,\nThe pro-\ud835\udc5d topology of a free group and algorithmic problems in semigroups,\nInternat. J. Algebra Comput. 4 (1994), no. 3, 359\u2013374.","DOI":"10.1142\/S021819679400004X"}],"container-title":["Journal of Group Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jgth-2023-0081\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jgth-2023-0081\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T08:55:43Z","timestamp":1756716943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jgth-2023-0081\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,28]]},"references-count":13,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,3,28]]},"published-print":{"date-parts":[[2025,9,1]]}},"alternative-id":["10.1515\/jgth-2023-0081"],"URL":"https:\/\/doi.org\/10.1515\/jgth-2023-0081","relation":{},"ISSN":["1433-5883","1435-4446"],"issn-type":[{"type":"print","value":"1433-5883"},{"type":"electronic","value":"1435-4446"}],"subject":[],"published":{"date-parts":[[2025,3,28]]}}}