Exercise 5, § 7

(a) Let $\Gamma$ be a compact subgroup of $G = \mathbf{GL}(n, \mathbf{Q}_p)$. Show that there exists a conjugate of $\Gamma$ which is contained in $\mathbf{GL}(n, \mathbf{Z}_p)$. (If $T$ is a lattice of $\mathbf{Q}_p^n$ with respect to $\mathbf{Z}_p$ (Commutative Algebra, Chapter VII, § 4, Definition 1), show that the stabilizer of $T$ in $\Gamma$ is open in $\Gamma$ and hence of finite index and $\sum_{\gamma \in \Gamma} \gamma T$ is a lattice which is stable under $\Gamma$.)

(b) Deduce that $G_f$ is the union of the conjugates of $\mathbf{GL}(n, \mathbf{Z}_p)$.

(c) Deduce that every finite subgroup $\Phi$ of $\mathbf{GL}(n, \mathbf{Q}_p)$ has order a divisor of $a_n(p)$, where $a_n(p)$ is defined by

(Using (a), it can be assumed that $\Phi \subset \mathbf{GL}(n, \mathbf{Z}_p)$. Then use Exercise 4.)

(d) Show that every finite subgroup of $\mathbf{SL}(n, \mathbf{Q}_p)$ has order a divisor of $s_n(p)$, where $s_n(p) = \frac{a_n(p)}{p-1}$ for $p \neq 2$ and $s_n(2) = \frac{a_n(2)}{2}$.