A note on Fuchs' Problem 34

Authors:
U. F. Albrecht and H. P. Goeters

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1319-1328

MSC (1991):
Primary 20K15, 20K30; Secondary 20J05

MathSciNet review:
1326993

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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate to what extent an abelian group is determined by the homomorphism groups where is chosen from a set of abelian groups. In particular, we address Problem 34 in Professor Fuchs' book which asks if can be chosen in such a way that the homomorphism groups determine up to isomorphism. We show that there is a negative answer to this question. On the other hand, there is a set which determines the torsion-free groups of finite rank up to quasi-isomorphism.

**1.**R. A. Beaumont and R. S. Pierce,*Torsion free groups of rank two*, Mem. Amer. Math. Soc. No.**38**(1961), 41. MR**0130297****2.**R. A. Beaumont and R. J. Wisner,*Rings with additive group which is a torsion-free group of rank two.*, Acta Sci. Math. Szeged**20**(1959), 105–116. MR**0106921****3.**Theodore G. Faticoni and Pat Goeters,*On torsion-free 𝐸𝑥𝑡*, Comm. Algebra**16**(1988), no. 9, 1853–1876. MR**952214**, 10.1080/00927878808823664**4.**László Fuchs,*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673**

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Additional Information

**U. F. Albrecht**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
albreuf@mail.auburn.edu

**H. P. Goeters**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
goetehp@mail.auburn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03324-2

Keywords:
Homomorphism group,
Martin's Axiom,
$p$-group,
torsion-free group

Received by editor(s):
October 1, 1993

Received by editor(s) in revised form:
December 12, 1993

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1996
American Mathematical Society