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Infinite rank Butler groups


Authors: Manfred Dugas and K. M. Rangaswamy
Journal: Trans. Amer. Math. Soc. 305 (1988), 129-142
MSC: Primary 20K20; Secondary 20K35, 20K40
DOI: https://doi.org/10.1090/S0002-9947-1988-0920150-X
MathSciNet review: 920150
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Abstract: A torsion-free abelian group $ G$ is said to be a Butler group if $ \operatorname{Bext} (G,\,T)$ for all torsion groups $ T$. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group $ G$ satisfies the T.E.P. over a pure subgroup $ H$ if and only if $ H$ is decent in $ G$ in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called $ {B_2}$-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a $ {B_2}$-group. We show under $ (V = L)$ that this is indeed the case for Butler groups of rank $ {\aleph _1}$. On the other hand it is shown that, under ZFC, it is undecidable whether a group $ B$ for which $ \operatorname{Bext} (B,\,T) = 0$ for all countable torsion groups $ T$ is indeed a $ {B_2}$-group.


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  • [1] U. Albrecht and P. Hill, Butler groups of infinite rank and Axiom $ 3$, Preprint.
  • [2] D. Arnold, Pure subgroups of finite rank completely decomposable groups, Proc. Abelian Group Theory Conference, Oberwolfach, Lecture Notes in Math., vol. 874, Springer-Verlag, 1981, pp. 1-31. MR 645913 (83j:20060)
  • [3] D. Arnold and C. Vinsonhaler, Pure subgroups of finite rank completely decomposable groups. II, Proc. Abelian Group Theory Conference, Honolulu, Lecture Notes in Math., vol. 1006, Springer-Verlag, 1983, pp. 97-143. MR 722614 (85g:20072)
  • [4] D. Arnold, Notes on Butler groups and balanced extensions, Preprint. MR 850285 (87h:20101)
  • [5] R. Baer, Abelian groups without elements of finite order, Duke Math J. 3 (1937), 68-122. MR 1545974
  • [6] L. Bican, Splitting in abelian groups, Czechoslovak Math. J. 28 (1978), 356-364. MR 0480778 (58:928)
  • [7] -, Purely finitely generated groups, Comment. Math. Univ. Carolin. 21 (1980), 209-218. MR 580678 (81i:20067)
  • [8] L. Bican and L. Salce, Infinite rank Butler groups, Proc. Abelian Group Theory Conference, Honolulu, Lecture Notes in Math., vol. 1006, Springer-Verlag, 1983, pp. 171-189. MR 722617 (86c:20050)
  • [9] L. Bican, L. Salcee, and J. Stepan, A characterization of countable Butler groups, Preprint. MR 818715 (87e:20100)
  • [10] M. C. R. Butler, A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc. 15 (1965), 680-698. MR 0218446 (36:1532)
  • [11] P. Eklof, Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207-225. MR 0453520 (56:11782)
  • [12] -, Applications of logic to the problem of splitting abelian groups, Logic Colloquium 76, North-Holland, 1977, pp. 287-299. MR 0540012 (58:27458)
  • [13] L. Fuchs, Infinite Abelian groups, vols. I and II, Academic Press, New York, 1971 and 1973. MR 0255673 (41:333)
  • [14] P. Griffith, A solution to the splitting mixed group problem of Baer, Trans. Amer. Math. Soc. 139 (1969), 261-269. MR 0238957 (39:317)
  • [15] R. Hunter, Balanced subgroups of abelian groups, Trans. Amer. Math. Soc. 215 (1976), 81-98. MR 0507068 (58:22337)
  • [16] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308. MR 0309729 (46:8834)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0920150-X
Keywords: Torsion-free abelian groups, Butler groups, pure subgroups
Article copyright: © Copyright 1988 American Mathematical Society

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