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Oct 21, 2015 at 21:19 comment added John Binder Yeah, after I saw @David's proof I agree.
Oct 21, 2015 at 19:57 comment added LSpice I think the fact that every compact subgroup of $\operatorname{GL}_n(\mathbb Q_p)$ is conjugate into $\operatorname{GL}_n(\mathbb Z_p)$ is not necessarily easier than the proof of the desired fact given by @DavidSpeyer above.
Oct 20, 2015 at 2:36 history answered John Binder CC BY-SA 3.0