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The structure of groups which are almost the direct sum of countable abelian groups


Author: Alan H. Mekler
Journal: Trans. Amer. Math. Soc. 303 (1987), 145-160
MSC: Primary 20K20; Secondary 03E35, 03E75, 20K25
DOI: https://doi.org/10.1090/S0002-9947-1987-0896012-2
MathSciNet review: 896012
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Abstract: The notion of being in standard form is defined for the groups described in the title of the paper which are of cardinality $ {\omega _1}$. Being in "standard form" is a structural description of the group. The consequences of being in standard form are explored, sometimes with the use of additional set-theoretic axioms. It is shown that it is consistent that a large class of these groups, including every weakly $ {\omega _1}$-separable $ {\omega _1}$-$ \Sigma $-cyclic group of cardinality $ {\omega _1}$, can be put in standard form.


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

  • [P] K. Devlin, A Yorkshire man's guide to proper forcing, Surveys in Set Theory, London Math. Soc. Lecture Notes No. 87, Cambridge Univ. Press, 1983, pp. 60-115. MR 823776 (87h:03081)
  • [E1] P. Eklof, Infinitary equivalence of abelian groups, Fund. Math. 81 (1974), 305-314. MR 0354349 (50:6829)
  • [E2] -, Set theoretic methods in homological algebra and Abelian groups, Presses Univ. Montréal, 1980. MR 565449 (81j:20004)
  • [E3] -, The structure of $ {\omega _1}$ -separable groups, Trans. Amer. Math. Soc. 279 (1983), 497-523. MR 709565 (84k:03124)
  • [EM1] P. Eklof and A. Mekler, On constructing indecomposable groups in $ L$, J. Algebra 49 (1977), 96-103. MR 0457197 (56:15412)
  • [EM2] -, On endomorphism rings of $ {\omega _1}$ -separable primary groups, Lecture Notes in Math., vol. 1006, Springer-Verlag, Berlin and New York, 1983, pp. 320-339.
  • [F] L. Fuchs, Abelian groups, Hungarian Academy of Sciences, Budapest, 1958. MR 0106942 (21:5672)
  • [F2] -, On $ {p^{\omega + 1}}$ -projective Abelian $ p$-groups, Publ. Math. Debrecen. 23 (1976), 309-313. MR 0427496 (55:528)
  • [FI] L. Fuchs and J. Irwin, On $ {p^{\omega + 1}}$ -projective $ p$-groups, Proc. London Math. Soc. 30 (1975), 459-470. MR 0374288 (51:10488)
  • [H] P. Hill, On the decomposition of groups, Canad. J. Math. 21 (1969), 762-768. MR 0249507 (40:2752)
  • [Meg 1] C. Megibben, Crawley's problem on the unique $ {\omega _1}$-elongation of $ p$-group is undecidable, Pacific J. Math. 107 (1983), 205-212. MR 701817 (84m:20058)
  • [Meg 2] -, $ {\omega _1}$-separable $ p$-groups (preprint).
  • [M1] A. Mekler, How to construct almost free groups, Canad. J. Math. 32 (1980), 1206-1228. MR 596105 (82b:20038)
  • [M2] -, Shelah's Whitehead groups and $ CH$, Rocky Mountain J. Math. 12 (1982), 272-278.
  • [M3] -, Proper forcing and Abelian groups, Abelian Group Theory, Lecture Notes in Math., vol. 1006, Springer-Verlag, Berlin and New York, 1983, pp. 285-303. MR 722625 (85h:03053)
  • [M4] -, c.c.c. forcing without combinatorics, J. Symbolic Logic 49 (1984), 830-832. MR 758934 (86f:03087)
  • [S1] S. Shelah, Infinite Abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243-256. MR 0357114 (50:9582)
  • [S2] -, Proper forcing, Lecture Notes in Math., vol. 940, Springer-Verlag, Berlin and New York, 1982. MR 675955 (84h:03002)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0896012-2
Article copyright: © Copyright 1987 American Mathematical Society

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