# A question about nilpotent group

If G is a finite nilpotent group, then every minimal normal subgroup of $G$ is contained in the center of $G$ and has prime order.

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Have you considered what $N\cap Z(G)$ might look like if $N$ were a minimal normal subgroup? – user641 Nov 8 '10 at 16:45
First, the minimal normal subgroup is a abelian p-group, consider the intersection of the minimal normal subgroup and the commutator subgroup of G is the identity. – Yuan Nov 8 '10 at 16:51

A finite nilpotent group is a product of $p$-groups. So you can do a very quick computation to show that you can reduce to the case where $G$ is a $p$-group.