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On a Problem Related to Homomorphism Groups in the Theory of Abelian Groups

Abstract

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ≅ Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.

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References

  1. 1.

    L. Fuchs, Infinite Abelian Groups, V. 1, Academic Press (1970).

  2. 2.

    P. Hill, “Two problems of Fuchs concerning tor and hom,” J. Algebra, 19, 379–383 (1971).

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Correspondence to S. Ya. Grinshpon.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 31–34, 2011/12.

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Grinshpon, S.Y. On a Problem Related to Homomorphism Groups in the Theory of Abelian Groups. J Math Sci 197, 602–604 (2014). https://doi.org/10.1007/s10958-014-1741-2

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Keywords

  • Abelian Group
  • Direct Summand
  • Homomorphism Group
  • Cardinal Number
  • Negative Solution