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  "body": "\n## Summary\n\nFor a prime $p$, a pro-$p$ group is a [[profinite group|KA0YDB6QXE]] that is an inverse limit of finite [[p-groups|06QYRTX9KQ]].\nThe $p$-adic integers are the basic example.\nPro-$p$ groups occur as Sylow subgroups of profinite groups and as Galois groups of maximal $p$-extensions of fields.\nEvery $p$-adic analytic group has an open subgroup that is a pro-$p$ group.\nMuch of the theory of finite $p$-groups carries over to them.\n\n## Explanations\n\n### L1 human\n\nFix a prime number $p$, such as $2$ or $3$.\nA finite $p$-group is a finite group whose number of elements is a power of $p$.\nA pro-$p$ group is built from a compatible tower of such finite groups, much as the $p$-adic integers are built from the integers modulo $p$, $p^2$, $p^3$, and so on.\nEach finite quotient of a pro-$p$ group by an open normal subgroup has order a power of $p$.\n",
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