Groups of homomorphisms of abelian groups

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SUMMARY

The discussion focuses on the groups of homomorphisms ##Hom(\mathbb{Q}/\mathbb{Z},\mathbb{Q})## and ##Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})##, highlighting the properties of the abelian groups involved. It is established that ##\mathbb{Q}/\mathbb{Z}## is a finitely generated ##\mathbb{Z}##-module and a torsion group, while ##\mathbb{Q}## is not finitely generated as a ##\mathbb{Z}##-module and contains no elements of finite order. The discussion emphasizes the need to clarify the meaning of "describing the groups of homomorphisms" and suggests exploring the generators of ##\mathbb{Q}##, particularly the reciprocals of prime numbers.

PREREQUISITES
  • Understanding of group homomorphisms
  • Knowledge of abelian groups
  • Familiarity with torsion groups
  • Basic concepts of finitely generated modules over rings
NEXT STEPS
  • Study the structure of torsion groups in detail
  • Learn about finitely generated modules over the integers
  • Explore the properties of homomorphisms between abelian groups
  • Investigate the role of prime numbers as generators in ##\mathbb{Q}##
USEFUL FOR

Mathematicians, particularly those studying abstract algebra, group theory, and module theory, will benefit from this discussion. It is also valuable for students seeking to deepen their understanding of homomorphisms in the context of abelian groups.

JD_Shadowplay
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Hello everybody!
I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed...

What i have to do is to describe ##Hom(\mathbb{Q}/\mathbb{Z},\mathbb{Q})## and ##Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})##. I know that both ##\mathbb{Q}/\mathbb{Z}## and ##\mathbb{Q}## are abelian groups, ##\mathbb{Q}/\mathbb{Z}## as a ##\mathbb{Z}##-module is finitely generated, but ##\mathbb{Q}## as a ##\mathbb{Z}##-module is not finitely generated.
Can anybody help me with this problem? How is it meant "to describe the groups of homomorphisms"?
Thank you in advance!
 
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A couple of things that may help
(Since Z is not a group under multiplication, you must be referring to addition)

- Q/Z is a torsion group. That is: every element is of finite order. Q has no elements of finite order.

- Try to find a set of Z generators of Q. Do the reciprocals of the prime numbers work?BTW: Why do you think Q/Z is a finitely generated Z module?
 
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