{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T08:10:17Z","timestamp":1707034217557},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":6529,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1996,3]]},"abstract":"<jats:p>The aim of this paper is to describe (without proofs) an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for \u201cold-fashioned\u201d model theoretic algebra and \u201cnew\u201d examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and group theory (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for group theory). However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.<\/jats:p><jats:p><jats:bold>\u00a71. Some model theoretic considerations<\/jats:bold>. Denote by<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007940_inline1\" \/>the theory of torsion-free abelian groups in the language of groups \u2112<jats:sub><jats:italic>gp<\/jats:italic><\/jats:sub>. A finitely generated group<jats:italic>G<\/jats:italic>satisfies<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007940_inline1\" \/>iff<jats:italic>G<\/jats:italic>is isomorphic to a finite direct power of \u2124. It follows that<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007940_inline1\" \/>axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007940_inline2\" \/>the theory of nontrivial divisible torsion-free abelian groups.<\/jats:p>","DOI":"10.2307\/421048","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:08:40Z","timestamp":1146985720000},"page":"84-93","source":"Crossref","is-referenced-by-count":2,"title":["From \u201cMetabelian \u211a-Vector Spaces\u201d to New \u03c9-Stable Groups"],"prefix":"10.1017","volume":"2","author":[{"given":"Olivier","family":"Chapuis","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007940_ref016","volume-title":"Stability in model theory","author":"Lascar","year":"1987"},{"key":"S1079898600007940_ref013","volume-title":"Building models by 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