Exercise 4, § 7

Let $n \in \mathbf{N}^*$, $p$ be a prime number and $G$ the set of matrices belonging to $\mathbf{GL}(n, \mathbf{Z}_p)$ all of whose elements are congruent to 1 modulo $p$ if $p \neq 2$ (resp. modulo 4 if $p = 2$). Then $G$ is an open subgroup of $\mathbf{GL}(n, \mathbf{Z}_p)$.

(a) Show that $G$ has no element of finite order $\neq 1$.

(b) Show that every finite subgroup of $\mathbf{GL}(n, \mathbf{Z}_p)$ is isomorphic to a subgroup of $\mathbf{GL}(n, \mathbf{Z}/p\mathbf{Z})$ if $p \neq 2$ (resp. $\mathbf{GL}(n, \mathbf{Z}/4\mathbf{Z})$ if $p = 2$).