{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T01:58:14Z","timestamp":1697853494490},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10511,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,6]]},"abstract":"<jats:p>Nielsen [7] has proved that every subgroup of a free group of finite rank is free. The theorem was later strengthened by Schreier [8] by eliminating the finiteness restriction on the rank. Several proofs of this theorem (known as the Nielsen-Schreier theorem, henceforth denoted by NS) have appeared since Schreier's 1927 paper (see [1] and [2]). All proofs of NS use the axiom of choice (AC) and it is natural to ask whether NS is equivalent to AC. L\u00e4uchli has given a partial answer to this question by proving [6] that the negation of NS is consistent with ZFA (Zermelo-Fraenkel set theory weakened to permit the existence of atoms). By the Jech-Sochor embedding theorem (see [3] and [4]) ZFA can be replaced by ZF. Some form of AC, therefore, is needed to prove NS. The main purpose of this paper is to give a further answer to this question.<\/jats:p><jats:p>In \u00a72 we prove that NS implies ACf<jats:sub>fin<\/jats:sub> (the axiom of choice for sets of finite sets). In \u00a73 we show that a strengthened version of NS implies AC and in \u00a74 we give a partial list of open problems.<\/jats:p><jats:p>Let <jats:italic>y<\/jats:italic> be a set; \u2223<jats:italic>y<\/jats:italic>\u2223 denotes the cardinal number of <jats:italic>y<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200032795_inline1\" \/>(<jats:italic>y<\/jats:italic>) is the power set of <jats:italic>y<\/jats:italic>. If <jats:italic>p<\/jats:italic> is a permutation of <jats:italic>y<\/jats:italic> and <jats:italic>t<\/jats:italic> \u2208 <jats:italic>y<\/jats:italic>, the <jats:italic>p<\/jats:italic>-orbit of <jats:italic>t<\/jats:italic> is the set {<jats:italic>p<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>(<jats:italic>t<\/jats:italic>): <jats:italic>n<\/jats:italic> is an integer}. If<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200032795_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>we call <jats:italic>p<\/jats:italic> a cyclic permutation of <jats:italic>y<\/jats:italic>. If <jats:italic>f<\/jats:italic> is a function with domain <jats:italic>y<\/jats:italic> and <jats:italic>x<\/jats:italic> \u2286 <jats:italic>y<\/jats:italic>, <jats:italic>f<\/jats:italic>\u2033<jats:italic>x<\/jats:italic> denotes the set {<jats:italic>f<\/jats:italic>(<jats:italic>t<\/jats:italic>):<jats:italic>t<\/jats:italic> \u2208 <jats:italic>x<\/jats:italic>}. If <jats:italic>A<\/jats:italic> is a subset of a group (<jats:italic>G<\/jats:italic>, \u00b0) (sometimes (<jats:italic>G<\/jats:italic>, \u00b0) will be denoted by <jats:italic>G<\/jats:italic>) then <jats:italic>A<\/jats:italic><jats:sup>\u22121<\/jats:sup>  = {<jats:italic>x<\/jats:italic><jats:sup>\u22121<\/jats:sup>:<jats:italic>x<\/jats:italic> \u2208 <jats:italic>A<\/jats:italic>} and [<jats:italic>A<\/jats:italic>] denotes the subgroup of <jats:italic>G<\/jats:italic> generated by <jats:italic>A<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274234","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:13:30Z","timestamp":1146953610000},"page":"458-467","source":"Crossref","is-referenced-by-count":5,"title":["Subgroups of a free group and the axiom of choice"],"prefix":"10.1017","volume":"50","author":[{"given":"Paul E.","family":"Howard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200032795_ref004","first-page":"297","article-title":"On \u0398-model of set theory","volume":"14","author":"Jech","year":"1963","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences, S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200032795_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF02952517"},{"key":"S0022481200032795_ref007","first-page":"79","volume-title":"Matematisk Tidsskrift B","author":"Nielsen","year":"1921"},{"key":"S0022481200032795_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566957"},{"key":"S0022481200032795_ref009","volume-title":"Group theory","author":"Shenkman","year":"1975"},{"key":"S0022481200032795_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1950-0032643-4"},{"key":"S0022481200032795_ref005","volume-title":"The theory of groups","author":"Kurosh","year":"1955"},{"key":"S0022481200032795_ref003","volume-title":"The axiom of choice","author":"Jech","year":"1973"},{"key":"S0022481200032795_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02941011"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200032795","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T22:36:02Z","timestamp":1558564562000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200032795\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1985,6]]}},"alternative-id":["S0022481200032795"],"URL":"https:\/\/doi.org\/10.2307\/2274234","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,6]]}}}