TITLE: Where do $p$-adic numbers and $p$-Sylow theory both appear?
QUESTION [7 upvotes]: Both $p$-adic numbers and $p$-Sylow theory are by design "arithmetic" ways of "localizing," so it stands to reason they might be in cahoots in certain contexts. Are they?

REPLY [4 votes]: Here's one reasonably simple link:  In terms of Sylow theory, a nice home for generalization is the land of profinite groups, where you have directly analogous Sylow theorems (with directly analogous proofs).  Among profinite groups there is the canonical $\widehat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$ and free profinite group of rank 1.  The decomposition
$$
\widehat{\mathbb{Z}}\cong \prod_p \mathbb{Z}_p
$$
shows that the $p$-adic integers are simply the $p$-Sylow subgroup of this very canonical group.