Show that $G$ is nilpotent

Question

How do I show that $G$ is nilpotent given that if $G$ is polycyclic and $G$ is residually finite p-group?

Answer 1

Note that $H'H^p=\Phi(H)$. Thus you have $[G', G]\le \Phi(H) \le \Phi(G)$. That means $G/\Phi(G)$ is nilpotent; use this to show $G$ is nilpotent.

(Assuming $G$ is finite.)