# Article

 Title: Butler groups and Shelah's Singular Compactness (English) Author: Bican, Ladislav Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 37 Issue: 1 Year: 1996 Pages: 171-178 . Category: math . Summary: A torsion-free group is a $B_2$-group if and only if it has an axiom-3 family $\frak C$ of decent subgroups such that each member of $\frak C$ has such a family, too. Such a family is called $SL_{\aleph_0}$-family. Further, a version of Shelah's Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group $B$ in a prebalanced and TEP exact sequence $0 \to K \to C \to B \to 0$ is a $B_2$-group provided $K$ and $C$ are so. (English) Keyword: $B_1$-group Keyword: $B_2$-group Keyword: prebalanced subgroup Keyword: torsion extension property Keyword: decent subgroup Keyword: axiom-3 family MSC: 20K20 MSC: 20K27 idZBL: Zbl 0857.20037 idMR: MR1396169 . Date available: 2009-01-08T18:22:49Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118821 . Reference: [AH] Albrecht U., Hill P.: Butler groups of infinite rank and axiom 3.Czech. Math. J. 37 (1987), 293-309. Zbl 0628.20045, MR 0882600 Reference: [B1] Bican L.: Purely finitely generated groups.Comment. Math. Univ. Carolinae 21 (1980), 209-218. MR 0580678 Reference: [B2] Bican L.: Butler groups of infinite rank.Czech. Math. J. 44 (119) (1994), 67-79. Zbl 0812.20032, MR 1257937 Reference: [B3] Bican L.: On $B_2$-groups.Contemporary Math. 171 (1994), 13-19. MR 1293129 Reference: [B4] Bican L.: Families of preseparative subgroups.to appear. Zbl 0866.20043, MR 1415629 Reference: [BF] Bican L., Fuchs L.: Subgroups of Butler groups.Communications in Algebra 22 (1994), 1037-1047. Zbl 0802.20045, MR 1261020 Reference: [BS] Bican L., Salce L.: Infinite rank Butler groups.Proc. Abelian Group Theory Conference, Honolulu Lecture Notes in Math., Springer-Verlag 1006 (1983), 171-189. Reference: [B] Butler M.C.R.: A class of torsion-free abelian groups of finite rank.Proc. London Math. Soc. 15 (1965), 680-698. Zbl 0131.02501, MR 0218446 Reference: [DHR] Dugas M., Hill P., Rangaswamy K.M.: Infinite rank Butler groups II.Trans. Amer. Math. Soc. 320 (1990), 643-664. MR 0963246 Reference: [F1] Fuchs L.: Infinite Abelian Groups, vol. I and II.Academic Press New York (1973 and 1977). MR 0255673 Reference: [F2] Fuchs L.: Infinite rank Butler groups.preprint. Reference: [H] Hodges W.: In singular cardinality, locally free algebras are free.Algebra Universalis 12 (1981), 205-220. Zbl 0476.03039, MR 0608664 Reference: [R1] Rangaswamy K.M.: A homological characterization of abelian $B_2$-groups.Comment. Math. Univ. Carolinae 35 (1994), 627-631. Reference: [R2] Rangaswamy K.M.: A property of $B_2$-groups.Proc. Amer. Math. Soc. 121 (1994), 409-415. MR 1186993 .

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