# $N$ normal in a finite group $G$, $|N| = 5$ and $|G|$ odd. Why is $N \subseteq Z(G)$?

Suppose that $N$ is a normal subgroup of a finite group $G$. If $|N| = 5$ and $|G|$ is odd, why is $N$ contained in $Z(G)$, the center of $G$?

I know how to do this when $|N| = 2$ and $|G|$ is even, but am not sure what to do with this one. Sylow's theorems maybe?

Thank you guys!

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Hint: $G/C_{G}(N)$ is a subgroup of the automorphism group of $N$, and that is easy to determine.