FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 8, PAGES 31-34

On a problem related to homomorphism groups in the theory of Abelian groups

S. Ya. Grinshpon

Abstract

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In this paper, for any reduced Abelian group $A$ whose torsion-free rank is infinite, we construct a countable set $\$\; \backslash mathfrak\; A(A)\; \$$ 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 $\$\; \backslash mathfrak\; A(A)\; \$$ 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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Last modified: March 23, 2013