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  "body": "\n## Definition\n\nLet $(G_i)_{i \\in I}$ be an inverse system of finite groups indexed by a directed set $I$, each with the discrete topology.\nGive its inverse limit $\\varprojlim_{i \\in I} G_i$ the subspace topology of the product $\\prod_{i \\in I} G_i$.\nA *profinite group* is a topological group $G$ that is isomorphic, as a topological group, to such an inverse limit.\n\n## Remarks\n\nA topological group is profinite if and only if it is [[compact|0TVJZYB6SH]], [[Hausdorff|0S1FJ8VEX9]] and [[totally disconnected|0RJSBR9GW1]].\nThen $G$ is the inverse limit of its finite quotients $G/N$, where $N$ runs over the open normal subgroups.\n\nExample: the [[p-adic integers|V6TJAFZ4MD]] $\\Z_p = \\varprojlim_n \\Z/p^n\\Z$, with the reduction maps $\\Z/p^{n+1}\\Z \\to \\Z/p^n\\Z$.\nEach $\\Z/p^n\\Z$ is finite and discrete, so $\\Z_p$ is profinite.\n\nNon-example: $\\Z$ with the discrete topology is Hausdorff and totally disconnected but not compact, so it is not profinite.\nIts profinite completion $\\hat\\Z = \\varprojlim_n \\Z/n\\Z$ is profinite and contains $\\Z$ as a dense subgroup.\n\nEvery finite group with the discrete topology is profinite.\nClosed subgroups, quotients by closed normal subgroups, and arbitrary products of profinite groups are profinite.\n",
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