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This is unlikely a research level question... one that would be answered in a blink of an eye, is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received when posted on the sibling site, I guess I might try my chances here, risking an immediate closure of the thread (nope, not homework...)

Give an example of a finitely generated pro-p group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely generated as an abstract group , such that $\hat{H} \ncong G$.

$\color{grey}{\rm edit}$: link to the other question:

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You can take in the free pro-p group on x,y the subgroup generated by x^{p+1},y^{p+1} and xy^px^{-1} – Benjamin Steinberg Sep 18 '13 at 12:41
up vote 7 down vote accepted

Let $G = \mathbb{Z}_p$, and let $H$ be a densely embedded copy of $\mathbb{Z} \oplus \mathbb{Z}$, e.g., $(1,0) \mapsto 1$ and $(0,1) \mapsto \alpha$ for some irrational $\alpha$. Assuming $\hat{H}$ means the pro-$p$ completion of $H$, it is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$. This is not isomorphic to $G$, because it contains a closed infinite index subgroup isomorphic to $G$.

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