I am trying to show that the dihedral group of $10$ elements, $D_{5}$ is not nilpotent Now, I know of a result that says the center of $D_{n} = Z(D_{n}) = \{ e\}$ when $n$ is odd, but I suppose that I need to prove that for any group $G$, $Z(G)=\{e\} \, \implies \, G$ is not nilpotent.
I came across a result that says that if $G$ is a nilpotent group with $|G|>1$, then $|Z(G)|>1$. I think that the contrapositive of this statement is that if $|Z(G)|=1$, then $G$ is either not nilpotent, or $|G|=1$. Is this the correct contraposition? If so, then I believe this result is exactly what I am looking for. However, sometimes I get logical negation wrong, which is why I'm asking.
If this is not true, or if this is not how one shows that $D_{5}$ is not nilpotent, could you please tell me and let me know how I should show it?
Thanks.
A nilpotent group is the direct product of its Sylow subgroups. Show that your group is not.