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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.

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