We know from the theory that if $\mathbb{L}$ is a finite Galois extension of the local field $\mathbb{K}$ then the ramification group $G_1$ is a $p$-group where $p$ is the characteristic of the residue field of $\mathbb{L}$.
Furthermore $G_1$ is a normal subgroup of $I$ the inertia group of $\mathbb{L}$/$\mathbb{K}$.
Is it true that $G_1$ is the largest $p$-group contained in $I$ ? If it's true, can i use this to show that $G_1$ is the biggest $pro$-$p$-group contained in $I$ when the Galois extension $\mathbb{L}$/$\mathbb{K}$ is infinite?