Finite abelian groups are neither injective nor projective

Oct 30, 2024 9:17 PM

Let $A$ be a nonzero finite abelian group. Prove that:

Hints

If you want to be sneaky, you can use this characterization of projective modules and this characterization of injective modules over a PID.
If you would prefer a direct approach, consider the following strategy:
- Note that a direct summand of a projective/injective module is also projective/injective
- By the Fundamental Theorem of Finite Abelian Groups, $A$ has a direct summand of the form $Z_{p^{k}}$ for some prime $p$ and positive integer $k$
- By considering the short exact sequence below, you can show $Z_{p^{k}}$ is neither injective nor surjective:
  $0 \to p^{k} Z_{p^{2 k}} \to Z_{p^{2 k}} \to Z_{p^{k}} \to 0$