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  "body": "\n## Definition\n\nLet $p$ be a prime number.\nA *pro-$p$ group* is a [[profinite group|KA0YDB6QXE]] $G$ such that for every [[open|QYCFH68RD1]] [[normal subgroup|0CB0DX5BVV]] $N$ of $G$ the finite quotient $G/N$ is a [[p-group|06QYRTX9KQ]].\n\n## Remarks\n\nEquivalently, $G$ is isomorphic as a topological group to an inverse limit of finite $p$-groups.\nExample: the [[p-adic integers|V6TJAFZ4MD]] $\\Z_p = \\varprojlim \\Z/p^n\\Z$.\nIts open subgroups are the subgroups $p^n \\Z_p$, and each quotient $\\Z_p / p^n \\Z_p \\cong \\Z/p^n\\Z$ has order $p^n$.\nNon-example: the profinite completion $\\hat\\Z$ of $\\Z$ is profinite, but it has the open normal subgroup $6\\hat\\Z$ with quotient $\\Z/6\\Z$, which is not a $p$-group for any prime $p$.\nEvery finite $p$-group with the discrete topology is a pro-$p$ group.\n",
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