Projective and injective objects in the category of finitely generated abelian groups Is it true that projective objects in the category of finitely generated abelian groups are precisely direct sums of integers? 
What about injective objects in the category of finitely generated abelian groups? I think they are exactly $\mathbb{Z}_p$ for prime $p $'s. I am not sure, however. 
 A: Projective objects: yes, by much the same arguments as for projective
implies free in the category of all Abelian groups.
The group $\Bbb Z_p$ of $p$ elements is
is not injective in the category of finitely
generated projectives. Consider the exact sequence
$$0\to p\Bbb Z/p^2\Bbb Z\to \Bbb Z/p^2\Bbb Z\to \Bbb Z/p\Bbb Z\to0.$$
It doesn't split. The only injective object in this category is
the zero module.
A: The structure theorem on finitely generated abelian groups has the consequence that every nonzero such group is a direct sum of cyclic groups either infinite or with prime power order. Since summands of projective (injective) objects are projective (injective), it's sufficient to classify the projective and injective nonzero cyclic groups.
Clearly $\mathbb{Z}$ is projective. Also, $\mathbb{Z}$ is not injective, because $2\mathbb{Z}$ is isomorphic to $\mathbb{Z}$, but the embedding $2\mathbb{Z}\to\mathbb{Z}$ doesn't split.
Consider the cyclic group $\mathbb{Z}/p^n\mathbb{Z}$ with $n>0$ and $p$ a prime. Since the short exact sequence
$$
0\to p^n\mathbb{Z}/p^{2n}\mathbb{Z}
\to\mathbb{Z}/p^{2n}\mathbb{Z}
\to\mathbb{Z}/p^{n}\mathbb{Z}\to0
$$
doesn't split, we conclude that $\mathbb{Z}/p^{n}\mathbb{Z}$ is neither projective nor injective.
