I am looking for a hint for the following exercise:
Show that the category of finitely generated abelian groups does not have enough injectives.
What would be a good way to start proving this?
2026-03-25 11:13:23.1774437203
The category of finitely generated abelian groups does not have enough injectives
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Hint: There are actually no nontrivial injective objects at all. To prove this, you can use the classification of finitely generated abelian groups. Given any nontrivial finitely generated abelian group, you just need to find a way to include it into a larger group such that it is not a direct summand. (In fact, since a direct summand of an injective object is injective, you just have to check this for cyclic groups!)