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  "body": "\n## Statement\n\nLet $p$ be a prime number and let $G$ be a [[group|G5S1M9]] of order $p$.\nThen $G$ is [[cyclic|0KVGH0J5B3]], and each element of $G$ other than the identity generates $G$.\n\n## Remarks\n\nIf $g \\neq e$, then the order of $\\langle g \\rangle$ is greater than $1$ and divides $p$ by [[Lagrange's theorem|3C53TD]], so it is $p$.\nThus each group of order $p$ is isomorphic to $\\mathbb{Z}/p\\mathbb{Z}$.\nA group of prime order has no subgroups except $\\{e\\}$ and itself.\n\n## Explanations\n\n### L1 human\n\nWhen the size of a group is a prime, there is no room for proper subgroups other than the trivial one.\nAny element other than the identity generates a subgroup larger than $1$, so it generates everything.\nThus, for example, each group with $7$ elements is a clock with $7$ hours.\n",
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