Fix a prime $p$. Let $G$ be a group endowed with the pro-$p$ topology, and let $H$ be an open subgroup of $G$.
I am trying to prove that the induced topology on $H$ is the pro-$p$ topology of $H$.
It is enough to show that each normal subgroup $N$ of $H$ with $[H:N]$ a power of $p$ contains a subgroup $K$ which is normal in $G$ with $[G:K]$ a power of $p$.
Since $H$ is open, we know that $[G:Core_G(H)]$ is a power of $p$. So, I am trying to find the right $K$ inside $Core_G(H)$, but I am having a hard time.
