Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Butler groups of infinite rank. II


Authors: Manfred Dugas, Paul Hill and K. M. Rangaswamy
Journal: Trans. Amer. Math. Soc. 320 (1990), 643-664
MSC: Primary 20K20; Secondary 20K40
DOI: https://doi.org/10.1090/S0002-9947-1990-0963246-8
MathSciNet review: 963246
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A torsion-free abelian group $ G$ is called a Butler group if $ \operatorname{Bext} (G,T) = 0.$ for any torsion group $ T$. We show that every Butler group $ G$ of cardinality $ {\aleph _1}$ is a $ {B_2}$-group; i.e., $ G$ is a union of a smooth ascending chain of pure subgroups $ {G_\alpha }$ where $ {G_{\alpha + 1}} = {G_\alpha } + {B_\alpha },{B_\alpha }$ a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding $ {\aleph _\omega }$ is a $ {B_2}$-group. Moreover, we are able to prove that the derived functor $ {\operatorname{Bext} ^2}(A,T) = 0$ for any torsion group $ T$ and any torsion-free $ A$ with $ \vert A\vert \leqslant {\aleph _\omega }$. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality $ \leqslant {\aleph _\omega }$ are $ {B_2}$-groups.


References [Enhancements On Off] (What's this?)

  • [A] D. Arnold, Notes on Butler groups and balanced extensions, Boll. Un. Math. Ital. A (6) 5 (1986), 175-184. MR 850285 (87h:20101)
  • [AH] U. Albrecht and P. Hill, Butler groups of infinite rank and axiom $ 3$, Czechoslovak Math. J. 37 (112) (1987), 293-309. MR 882600 (88f:20070)
  • [B] L. Bican, Splitting in mixed groups, Czechoslovak Math. J. 28 (1978), 356-364. MR 0480778 (58:928)
  • [BS] L. Bican and L. Salce, Infinite rank Butler groups, Proc. of Abelian Group Theory Conference, Honolulu, Lecture Notes in Math., vol. 1006, Springer-Verlag, Berlin, 1983, pp. 171-189. MR 722617 (86c:20050)
  • [BSS] L. Bican, L. Salce and J. Stepan, A characterization of countable Butler groups, Rend. Sem. Mat. Univ. Padova 74 (1985), 51-58. MR 818715 (87e:20100)
  • [DR1] -, On torsion-free abelian $ k$-groups, Proc. Amer. Math. Soc. 99 (1987), 403-408. MR 875371 (88b:20086)
  • [DR2] -, Infinite rank Butler groups, Trans. Amer. Math. Soc. (to appear). MR 920150 (88i:20073)
  • [D] M. Dugas, On some subgroups of infinite rank Butler groups, Rend. Sem. Mat. Univ. Padova 79 (1988), 153-161. MR 964027 (89h:20079)
  • [E] P. Eklof, Set theoretic methods in homological algebra and abelian group, Presses Univ. Montreal, Montreal, 1980. MR 565449 (81j:20004)
  • [EF] P. Eklof and L. Fuchs, Baer modules over valuation domains, Ann. Mat. Pura Appl. 150 (1988), 363-373. MR 946041 (89g:13006)
  • [Fu] L. Fuchs, Infinite abelian groups, vols. I and II, Academic Press, New York, 1977 and 1973. MR 0255673 (41:333)
  • [FH] L. Fuchs and P. Hill, The balanced-projective dimension of abelian $ p$-groups, Trans. Amer. Math. Soc. 293 (1986), 99-112. MR 814915 (87a:20056)
  • [HM] P. Hill and C. Megibben, Torsion-free groups, Trans. Amer. Math. Soc. 295 (1986), 735-751. MR 833706 (87e:20102)
  • [H] W. Hodges, In singular cardinality, locally free algebras are free, Algebra Universalis 12 (1981), 205-220. MR 608664 (82i:08005)
  • [Hu] R. Hunter, Balanced subgroups of abelian groups, Trans. Amer. Math. Soc. 215 (1976), 81-98. MR 0507068 (58:22337)
  • [S] S. Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), 319-349. MR 0389579 (52:10410)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20K20, 20K40

Retrieve articles in all journals with MSC: 20K20, 20K40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0963246-8
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society