When does $\gamma_i(G/H) = 1$ imply $\gamma_i(G) < H$ ?

Question

Let $\gamma_i$ be the $i$th term in the lower central series for a group $G$ and let $H < G$ be a subgroup of G. When does $\gamma_i(G/H) = 1$ imply $\gamma_i(G) < H$ ?

Answer 1

You can prove by induction that $\gamma_i(G/H)=\gamma_i(G)H/H$. Then the result follows.

Proof:

$\gamma_1(G/H)=G/H=\gamma_1(G)H/H$.

$\gamma_{i+1}(G/H)=[\gamma_i(G/H),G/H]=[\gamma_i(G)H/H,G/H]$ is the group generated by elements $[g_1H,g_2H]=[g_1,g_2]H$ with $g_1\in \gamma_i(G)$ and $g_2\in G$. That is $\gamma_{i+1}(G/H)=[\gamma_i(G),G]H/H=\gamma_{i+1}(G)H/H$.