A nice little finite group exercise

Exercise 1: Using the Structure Theorem for Finite Abelian Groups, or otherwise, show that a Sylow $p$-subgroup of a finite abelian group is a characteristic subgroup.

Exercise 2: If $N \triangleleft G$ is a minimal nontrivial normal subgroup, and is finite and abelian, then $|N|$ is a power of a prime.