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Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices with 1's in the diagonal, for some $d$?.

This question is the analogous of this well known result: every finitely generated torsion free nilpotent group is isomorphic to a subgroup of $U_d(\mathbb{Z})$ for some $d$.

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It can be embedded in $GL_n(\mathbb{Z}_p)$, for some $n$ (such a group is $p$-adic analytic). – Yassine Guerboussa Aug 27 '15 at 21:36

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