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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!

BIS HD
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    Since $\rm{GL}_n(\mathbf{Z}_p)$ is an open (hence closed) subgroup of $\rm{GL}_n(\mathbf{Q}_p)$, any closed subgroup of the former is also a closed subgroup of the latter. Also, you seem to be actually interested in compact $p$-adic analytic groups. – Keenan Kidwell Nov 21 '13 at 13:11
  • Since I'm reading about Galois representations of the absolute Galois group $G_S(k)$ of a field $k$ that is unramified outside a finite set of primes $S$, respectively it's pro-$p$-factor groups, all the images of those groups under a continuous representation are in fact compact. (since they're profinite, pro-$p$-groups resp.) I want to know something about the $p$-adic analytic quotients of those groups. – BIS HD Nov 21 '13 at 13:27
  • Dear @BIS HD, I'm just pointing out that not every $p$-adic analytic group is compact, so not all such groups can be closed subgroups of some $\rm{GL}_n(\mathbf{Z}_p)$ (the Wikipedia article sort of makes it sound like every $p$-adic analytic group is a pro-$p$ group, which is not the case). – Keenan Kidwell Nov 21 '13 at 13:42
  • @KeenanKidwell That were my thoughts about the Wikipedia article, too. I guess it's clear from my answer that my statement is right for the class of pro-$p$ groups, but I have no idea if it stays true for general $p$-adic analytic groups. Do you have a counterexample? – BIS HD Nov 21 '13 at 15:10
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    Do you mean a counterexample to the statement that a $p$-adic analytic group is a closed subgroup of some $\rm{GL}_n(\mathbf{Z}_p)$? Or of $\rm{GL}_n(\mathbf{Q}_p)$? For the first statement, any non-compact group, such as a $\mathbf{Q}_p$ itself, will work. I don't know about the second. Certainly it's true for groups arising as the $\mathbf{Q}_p$-points of a linear algebraic group over $\mathbf{Q}_p$. – Keenan Kidwell Nov 21 '13 at 15:29
  • Hey guys... you know that you can edit the wikipedia article? If it's wrong and/or ambiguous, it would be great if you would fix it! Thanks. – Mathi Shard Jun 26 '17 at 16:13

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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 $n$.

BIS HD
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