Given two number fields $K$ and $L$, with $K\subset L$ and $L$ normal over $K$, we let $R$ and $S$ be their respective rings of algebraic integers. For a prime $P$ of $R$ and a prime $Q$ of $S$ lying over $P$, for every $m\in \mathbb{N}_0$ define $V_m=\left \{\sigma\in \text{Gal}\left (L/K\right ):\sigma \left (\alpha\right )\equiv \alpha\pmod{Q^{m+1}}\text{ }\forall \alpha\in S\right \}$.
Exercise $23$ on Marcus's Number Fields asks you to show that $V_1$ is the Sylow $p$-subgroup of $V_0$, where $p$ is the only prime in $\mathbb{N}$ lying under $Q$.
Now, if we define $E:=V_0$, one proves that $E/V_1$ is embedded in the multiplicative group $\left (S/Q\right )^{\times}$ and that for $m\geq 2$, $V_{m-1}/V_m$ is embedded in the additive group of $S/Q$. Also we know that $\left |E\right |$ is the ramification index of $Q$ over $P$.
Letting $m=\left [L:K\right ]$ and $n=\left [K:\mathbb{Q}\right ]$ we have that $E/V_1$ is cyclic of order dividing $p^{mn}-1$ since every finite subgroup of the multiplicative group of a field is cyclic. Also we obtain that $p\nmid \left |\frac{E}{V_1}\right |$. Therefore it only remains to prove that if $V_1$ is not trivial, then it is a $p$-group. How would you show that?
You should show that the $V_m$ are eventually trivial, say $V_N=\{1\}$ for some large $N$ (we know they must stabilize, since $\mathrm{Gal}(L/K)$ is finite, so just unpack the definition to see that the stable group $\bigcap_{m}V_m$ is trivial).
This gives us a filtration $\{1\}=V_N\trianglelefteq V_{N-1}\trianglelefteq \ldots \trianglelefteq V_2\trianglelefteq V_1$, so that $|V_1|=\prod_{i=1}^{N-1}[V_{i}:V_{i+1}]$. But we know that $V_i/V_{i+1}$ embeds into the additive group of $S/Q$ for $i\geq 1$, and since the additive group of $S/Q$ is a $p$-group, each $[V_i:V_{i+1}]$ is a power of $p$.