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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], and let $\\mathcal{N}$ be the set of [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]] in $G$.\nFor $N \\in \\mathcal{N}$ give the finite [[quotient group|0KYX2N69J0]] $G/N$ the discrete topology.\nThe *profinite completion* of $G$ is\n$$\\hat G = \\varprojlim_{N \\in \\mathcal{N}} G/N = \\Big\\{ (g_N N)_{N \\in \\mathcal{N}} \\in \\prod_{N \\in \\mathcal{N}} G/N : g_M N = g_N N \\text{ whenever } M \\subseteq N \\Big\\},$$\nwith the group law taken coordinatewise and the topology induced by the [[product topology|0GBG8B49AA]].\nThe map $\\iota \\colon G \\to \\hat G$, $\\iota(g) = (gN)_{N \\in \\mathcal{N}}$, is the canonical homomorphism.\n\n## Remarks\n\n$\\hat G$ is a [[profinite group|KA0YDB6QXE]], and $\\iota(G)$ is dense in $\\hat G$.\nThe kernel of $\\iota$ is the intersection of all $N \\in \\mathcal{N}$, so $\\iota$ is injective if and only if $G$ is [[residually finite|HEE922EHQ8]].\nExample: for $G = \\Z$, the finite-index subgroups are $n\\Z$ with $n \\ge 1$, and $\\hat\\Z = \\varprojlim \\Z/n\\Z$; by the Chinese remainder theorem it is isomorphic to $\\prod_p \\Z_p$.\nHere $\\iota$ is injective, since $\\bigcap_n n\\Z = \\set{0}$.\nNon-example of injectivity: $(\\Q, +)$ has no proper subgroup of finite index, so $\\hat\\Q$ is trivial and $\\iota$ is not injective.\nUniversal property: every homomorphism from $G$ to a profinite group $P$ factors uniquely through $\\iota$ by a continuous homomorphism $\\hat G \\to P$.\n",
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