{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T15:41:58Z","timestamp":1777563718724,"version":"3.51.4"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5490,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,3]]},"abstract":"<jats:p>In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc>1.<jats:italic>Let G be a perfect group of finite Morley rank and let C<\/jats:italic><jats:sub>0<\/jats:sub><jats:italic>be a definable central subgroup of G such that G\/C<jats:sub>0<\/jats:sub>is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C<\/jats:italic><jats:sub>0<\/jats:sub>= 1.<\/jats:p><jats:p>Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.<\/jats:p><jats:p>C<jats:sc>orollary<\/jats:sc>1.<jats:italic>Let G be a perfect group of finite Morley rank such that G\/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite<\/jats:italic>([4],<jats:italic>Section<\/jats:italic>27.5).<\/jats:p><jats:p>An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of<jats:italic>tame<\/jats:italic>simple<jats:italic>K*-groups<\/jats:italic>of finite Morley rank, which constitutes an approach to the Cherlin-Zil\u2019ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of<jats:italic>tameness<\/jats:italic>, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from<jats:italic>K<\/jats:italic>-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.<\/jats:p>","DOI":"10.2307\/2586751","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:02:25Z","timestamp":1146938545000},"page":"68-74","source":"Crossref","is-referenced-by-count":16,"title":["On central extensions of algebraic groups"],"prefix":"10.1017","volume":"64","author":[{"given":"Tuna","family":"Altinel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gregory","family":"Cherlin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013906_ref001","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1996.6950"},{"key":"S0022481200013906_ref009","volume-title":"Lectures on Chevalley groups","author":"Steinberg","year":"1967"},{"key":"S0022481200013906_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4314-4"},{"key":"S0022481200013906_ref007","volume-title":"Groupes stables","author":"Poizat","year":"1987"},{"key":"S0022481200013906_ref004","volume-title":"Linear algebraic groups","author":"Humphreys","year":"1981"},{"key":"S0022481200013906_ref006","volume-title":"An introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S0022481200013906_ref002","first-page":"349\u2013446","volume-title":"Algebraic K-theory II","author":"Bass","year":"1973"},{"key":"S0022481200013906_ref003","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534457.001.0001","volume-title":"Groups of finite Morley rank","author":"Borovik","year":"1994"},{"key":"S0022481200013906_ref005","doi-asserted-by":"crossref","first-page":"1\u201325","DOI":"10.4064\/fm-71-1-1-25","article-title":"On \u03c91-categorical theories of fields","volume":"71","author":"Macintyre","year":"1971","journal-title":"Fundamenta Mathematicae"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013906","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:19:37Z","timestamp":1707031177000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013906\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1999,3]]}},"alternative-id":["S0022481200013906"],"URL":"https:\/\/doi.org\/10.2307\/2586751","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,3]]}}}