Huber, Martin A simple proof for a theorem of Chase. (English) Zbl 0583.20042 Rend. Sem. Mat. Univ. Padova 74, 45-49 (1985). The author gives a new proof of the following well-known Theorem of Chase: Assume \(2^{\aleph_ 0}<2^{\aleph_ 1}\). If G is a torsion- free group such that Ext(G,Z) is torsion, then G is strongly \(\aleph_ 1\)-free. The idea of the proof is obvious for some one familiar with recent structure theorems of Ext, the solution of the Whitehead problem etc: Use the weak diamond (Devlin and Shelah) which follows from the set theoretic assumption. The smoothest way to prove Chase’s theorem is then by Baer’s factor systems. Reviewer: R.Göbel Cited in 1 Document MSC: 20K20 Torsion-free groups, infinite rank 20K27 Subgroups of abelian groups 20K35 Extensions of abelian groups Keywords:Theorem of Chase; torsion-free group; strongly \(\aleph _ 1\)-free; Ext; Whitehead problem; weak diamond PDFBibTeX XMLCite \textit{M. Huber}, Rend. Semin. Mat. Univ. Padova 74, 45--49 (1985; Zbl 0583.20042) Full Text: Numdam EuDML References: [1] S. Chase , On group extensions and a problem of J. H. C. Whitehead , in Topics in Abelian Groups , pp. 173 - 197 , Chicago : Scott, Foresman and Co. , 1963 . MR 169881 · Zbl 0171.28703 [2] K. Devlin - S. Shelah , A weak version of \? which follows from 2N0 < 2N1 , Israel J. Math. , 29 ( 1978 ), pp. 239 - 247 . Zbl 0403.03040 · Zbl 0403.03040 · doi:10.1007/BF02762012 [3] P. Eklof - M. Huber , On the rank of Ext , Math. Z. , 174 ( 1980 ), pp. 159 - 185 . Article | MR 592913 | Zbl 0424.20051 · Zbl 0424.20051 · doi:10.1007/BF01293536 [4] L. Fuchs , Infinite Abelian Groups , Vol. I , London , New York , Academic Press , 1970 . MR 255673 | Zbl 0209.05503 · Zbl 0209.05503 [5] H. Hiller - M. Huber - S. Shelah , The structure of Ext(A, Z) and V = L , Math. Z. , 162 ( 1978 ), pp. 39 - 50 . Article | MR 492007 | Zbl 0366.20041 · Zbl 0366.20041 · doi:10.1007/BF01437821 [6] S. Shelah , Infinite abelian groups. Whitehead problem and some constructions , Israel J. Math. , 18 ( 1974 ), pp. 243 - 256 . MR 357114 | Zbl 0318.02053 · Zbl 0318.02053 · doi:10.1007/BF02757281 [7] S. Shelah , A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals , Israel J. Math. , 21 ( 1975 ), pp. 319 - 349 . MR 389579 | Zbl 0369.02034 · Zbl 0369.02034 · doi:10.1007/BF02757993 [8] S. Shelah , On uncountable abelian groups , Israel J. Math. , 32 ( 1979 ), pp. 311 - 330 . MR 571086 | Zbl 0412.20047 · Zbl 0412.20047 · doi:10.1007/BF02760461 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.