I am reading some lecture notes and came across the fact that the group $G_p \leq G_{\mathbb{Q}}$ (where $G_{\mathbb{Q}}$ is the absolute Galois Group of the rationals and $G_p$ is the decomposition group corresponding to the rational prime $p$) satisfies
$G_p \cong G_{\mathbb{Q}_p}$
where $G_{\mathbb{Q}_p}$ is the absolute Galois Group of the $p$-adics.
I cannot figure out why this is true. I have a vague intuition that if an element of $G_{\mathbb{Q}}$ fixes $p$ it should somehow fix the $p$-adic distance, but I am not sure how to formalize this thought.
Any help or reference would be much appreciated.