{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:20:33Z","timestamp":1775463633817,"version":"3.50.1"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12520,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,12]]},"abstract":"<jats:p>Let <jats:italic>M<\/jats:italic> be a countable algebraically closed group, \u03ba an uncountable cardinal. We will prove in this paper the following theorems.<\/jats:p><jats:p>Theorem 1. <jats:italic>There is an algebraically closed group N of cardinality \u03ba which is \u221e \u2013 \u03c9-equivalent to M<\/jats:italic>.<\/jats:p><jats:p>Theorem 2. <jats:italic>There is an algebraically closed group N of cardinality \u03ba which is \u221e \u2013 \u03c9-equivalent to M, and contains a free abelian group of cardinality \u03ba<\/jats:italic>.<\/jats:p><jats:p>Theorem 3. <jats:italic>There are 2<jats:sup>\u03ba<\/jats:sup> nonisomorphic algebraically closed groups of cardinality \u03ba which are \u221e \u2013 \u03c9-equivalent to M<\/jats:italic>.<\/jats:p><jats:p>Theorem 4. <jats:italic>There is an algebraically closed group N of cardinality \u03ba which is \u221e \u2013 \u03c9-equivalent to M and satisfies: Every subgroup of N of uncountable reqular cardinality contains a free subgroup of the same cardinality<\/jats:italic>.<\/jats:p><jats:p>Theorems 2 and 4 illustrate Theorem 3 by exhibiting two groups <jats:italic>N<\/jats:italic> \u2261 <jats:sub>\u221e\u03c9<\/jats:sub><jats:italic>M<\/jats:italic> of cardinality \u03ba which are nonisomorphic by obvious reasons. We state and prove Theorem 1 separately in order to give an easy example of our principal tool: the use of automorphisms instead of indiscernibles (see \u00a72).<\/jats:p>","DOI":"10.2307\/2273291","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:51:21Z","timestamp":1146937881000},"page":"522-532","source":"Crossref","is-referenced-by-count":8,"title":["Algebraically closed groups of large cardinality"],"prefix":"10.1017","volume":"44","author":[{"given":"Saharon","family":"Shelah","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Ziegler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048404_ref009","volume-title":"Bulletin of the Greek Mathematical Society","author":"Shelah"},{"key":"S0022481200048404_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1970894"},{"key":"S0022481200048404_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1978-0480750-4"},{"key":"S0022481200048404_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-34.4.465"},{"key":"S0022481200048404_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1970-0260879-9"},{"key":"S0022481200048404_ref002","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-11445"},{"key":"S0022481200048404_ref005","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.41.247"},{"key":"S0022481200048404_ref010","volume-title":"Proceedings of the Logic Workshop, Berlin, July 1977, Archiv f\u00fcr Mathematische Logik","author":"Shelah"},{"key":"S0022481200048404_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757000"},{"key":"S0022481200048404_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(76)90150-2"},{"key":"S0022481200048404_ref003","volume-title":"Proceedings of a Conference on Word and Decision Problems, Oxford, 1976","author":"Ziegler","year":"1979"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048404","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T16:27:59Z","timestamp":1558888079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048404\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,12]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1979,12]]}},"alternative-id":["S0022481200048404"],"URL":"https:\/\/doi.org\/10.2307\/2273291","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,12]]}}}