Let $L/K$ be a finite Galois extension of valued fields with unique extension of the valuation from $K$ and $G_s$ be the $s$-th ramification group. One can show that $G_0/G_1 \hookrightarrow \lambda^{\times}$ where $\lambda$ is the residue field of $L$.
On page 177 in Neukirch's book, he writes that this implies that $G_1$ is the unique $p$-Sylow subgroup of $G_0$ if $p = \operatorname{char}{\lambda}$.
Why? If it were a $p$-Sylow subgroup, then it would be unique by normality. By the isomorphism we also find that it suffices to show that $G_1$ is a $p$-group. But why is it a $p$-group?
To expand a little on Lubin's comment:
On page 177, Neukirch shows $G_s/G_{s+1} \hookrightarrow \lambda^+$ for $s \geq 1$ as well. So $G_s/G_{s+1}$ is a $p$-group. Because $\operatorname{Gal}(L/K)$ is finite, the normal chain eventually stops in $G_n = \{\operatorname{id}\}$. So we find that $$|G_1| = |G_1/G_n| = (G_1 : G_2) (G_2 : G_3) \dots (G_{n-1} : G_n)$$ is a power of $p$ because $G_1/G_2, \dots, G_{n-1}/G_n$ are all $p$-groups.