I am working through Paolo Aluffi's new GSM text on my own (self-study).
On page 63, he asks the reader to
Prove that $\Bbb{Q}$ is not the direct product of two nontrivial groups.
For some context, this is an exercise following a section entitled "The category Grp". I am assuming that he means "is not isomorphic to the direct product of two nontrivial groups", and I can see two possible ways to proceed with this proof, but have been unsuccessful with either approach.
Approach 1: Show that the additive group of rationals has a property that is preserved by isomorphism that the direct product of two nontrivial groups does not have or vice-versa. This seems challenging unless I can significantly narrow down the properties that a direct product of two nontrivial groups that was isomorphic to $\Bbb{Q}$ would necessarily have.
Approach 2: Considering the section in which this question occurs, show that if $G$ and $H$ are nontrivial groups and $\Bbb{Q} \cong G \times H$, then there are homomorphisms $\varphi_{G}:\Bbb{Q} \rightarrow G$ and $\varphi_{H}:\Bbb{Q} \rightarrow H$ which do not factor or do not factor uniquely through the product $G \times H$. This would be a contradiction, as $G \times H$ is a final object in the category Grp.
I would greatly appreciate suggestions on how to proceed further with either of these approaches or with alternate approaches.
Best Answer
The endomorphism ring of a direct product is never a domain, yet the endomorphism ring of ℚ is a field.