# Groups of homomorphisms of abelian groups

1. Oct 10, 2014

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"?

2. Oct 12, 2014

### lavinia

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?

Last edited: Oct 12, 2014