Problem: If $N$ is a normal subgroup of order $p$ where $p$ is the smallest prime dividing the order of a finite group $G$, then $N$ is in the center of $G$.
Solution: Since $N$ is normal, we can choose for $G$ to act on $N$ by conjugation. This implies that there is a homomorphism from $G$ to the automorphism group of $N$, which has $p - 1$ elements. Thus the homomorphism is trivial and $N$ is in the center of $G$
My first question is why conjugation implies the automorphism group.
My second question is why the automorphism only has $p-1$ elements; i.e. why is conjugation by the identity excluded even though it's a valid conjugation.