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  "body": "\n## Definition\n\nLet $p$ be a prime number.\nA [[group|G5S1M9]] $G$ is a $p$-group when, for each $g \\in G$, the [[order|0WE2NHZJ7B]] of $g$ is $p^n$ for some integer $n \\geq 0$.\n\n## Remarks\n\nA finite group is a $p$-group if and only if $|G| = p^k$ for some $k \\geq 0$.\nOne direction uses [[Lagrange's theorem|3C53TD]], and the other direction uses [[Cauchy's theorem|0JM7PA288B]].\nThe [[dihedral group|0JZ27DMJ5D]] $D_4$ of order $8$ is a $2$-group that is not abelian.\nThe trivial group is a $p$-group for each prime $p$.\n\n## Explanations\n\n### L1 human\n\nIn a $p$-group, each element comes back to the identity after a power of $p$ steps.\nFor finite groups this means that the number of elements is a power of $p$.\nThese groups are the building blocks of the Sylow theory.\n",
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