If $H \leq G$, $N \lhd G$, G=HN', then $G = H( \gamma _i N)$ for all $i$.
Here, $\gamma_iN$ are the terms in the lower central series of $N$, i.e., $\gamma_1 = N$ and $\gamma_{i+1}N = [\gamma_i N, N] $.
There is a hint:
N =(H \cap N) N'.
I have no idea what to do with the hint, and how to get a proof. I am wondering if I can get some help.
Thank you very much.
[This is on page 128 of A Course in the Theory of Groups (GTM 80) written by Derek J.S. Robinson.]