How do I show that $G$ is nilpotent given that if $G$ is polycyclic and $G$ is residually finite p-group?
closed as off-topic by Derek Holt, colormegone, user26857, zz20s, Michael Albanese Mar 12 '16 at 21:53
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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.)