$p$-adic analytic group are closed subgroups of $GL_n(\mathbb{Z}_p)$ for some $n$

Question

The article on pro-$p$-groups on Wikipedia tells us, that any $p$-adic analytic group can be found as a closed subgroup of $GL_n(\mathbb{Z}_p)$ for some $n \geq 0$. Do you have a reference for that fact? In the book "Analytic Pro-$p$ Groups" of Dixon et. al. I can't find that fact, although it must have something to do with the manifold structure of the $p$-adic analytic group.

Also is our $p$-adic analytic group a closed subgroup of $GL_n(\mathbb{Q}_p)$ as well?

Thank you in advance!

Answer 1

The answer is a bit hidden in the book and in addition, Wikipedia is not very clear about it: In the interlude A (p. 97), it is written

A pro-$p$ group $G$ is of finite rank iff. $G$ is isomorphic to a closed subgroup of $GL_n(\Bbb Z_p)$ for some $n$.

Also, it holds

A pro-$p$-group $G$ is of finite rank iff. $G$ is a $p$-adic analytic group.

Hence we can deduce:

A pro-$p$-group $G$ is $p$-adic analytic iff. $G$ is isomorphic to a closed subgroup of $GL_n(\Bbb Z_p)$ for some $n$.