Suppose $G$ is a nonabelian nilpotent group and let $x \in G$. I am trying to show that $\langle [G,G], x \rangle$ is a proper subgroup of $G$. If I can show that the nilpotency class of $\langle [G,G],x \rangle$ is less than the nilpotency class of $G$, why would this imply that the subgroup is proper?
To show that the nilpotency class of $\langle [G, G], x \rangle$ is less than that of $G$, I'm confused as to whether I should be looking at a central series or lower central series to proceed. I would appreciate any direction on how to approach this.
Let $H=\langle[G,G],x\rangle$ and let $$ 1=Z_0(G)<Z_1(G)<\ldots<Z_s(G)=G $$ be an upper central series of $G$. Since $$ [G,G]\leq Z_{s-1}(G)\cap H\leq Z_{s-1}(H) $$ then $H/Z_{s-1}(H)$ is a cyclic group. And so $Z_{s-1}(H)=H$.