I don't believe the following statement is true in general
A normal subgroup of prime order is central.
But then what is wrong with my logic here?
Suppose $N \triangleleft G$ with $|N| = p$ prime. Since $N$ is closed under conjugation by elements of $G$, it is a union of conjugacy classes. Since it contains the identity, its contribution to the class equation of $G$ is some partition of $p$ containing at least one $1$. The number of ones appearing in this partition is the size of the intersection of the center of $G$ and $N$, which must be a subgroup of $N$. The only possibility is $1+\ldots+1$ ($p$ ones), meaning all five elements of $N$ are in the center of $G$. Therefore $N$ is in the center of $G$.