Suppose $G$ has a lower central series and $H$ is a proper subgroup of $G$. Show an index $k$ such that $G^{k+1} \subset H$ but $G^k \not\subset H$.

Question

Suppose $G$ has a lower central series and $H$ is a proper subgroup of $G$. Show that there is an index $k$ such that $G^{k+1} \subset H$ but $G^k \not\subset H$.

If $G$ has a lower central series then, $$G=G^{0} \triangleright G^{1} \triangleright \cdots \triangleright G^{n}=\{e\}$$ is a subnormal series where $G^i=[G^{i-1},G]=\langle [x,y]=xyx^{-1}y^{-1} \mid x \in G^{i-1}, y \in G \rangle$

I'm not sure as to why should the claim be true. Any ideas?

Answer 1

$G^n=\{e\}\subset H$. And $G^0=G\not\subset H$, since $H$ is a proper subgroup. So let $k=\operatorname{min}\{j\mid G^j\subset H\}-1$.