# 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.

-
What have you tried so far? –  Qiaochu Yuan Nov 8 '10 at 16:37
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

## 1 Answer

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.

Then look at this question to answer Steve D's comment query.

-
Well, we can get by with more generality by considering that eventually [G,G,...,N] must be central, and contained in N. This shows the question is true even if we remove the word "finite". –  user641 Nov 8 '10 at 17:25
@Steve D: Agreed. –  Arturo Magidin Nov 8 '10 at 17:26