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  "body": "\n## Statement\n\nLet $p$ be a prime number.\nThen the group $(\\mathbb{Z}/p\\mathbb{Z})^\\times$ is [[cyclic|0KVGH0J5B3]] of order $p - 1$.\n\n## Remarks\n\nSo $(\\mathbb{Z}/p\\mathbb{Z})^\\times$ is isomorphic to the additive group $\\mathbb{Z}/(p-1)\\mathbb{Z}$, but no simple general formula for a generator is known.\nA generator is a primitive root modulo $p$, and there are $\\varphi(p - 1)$ of them.\nFor $p = 7$, the powers of $3$ modulo $7$ are $1, 3, 2, 6, 4, 5$, so $3$ is a primitive root modulo $7$.\nMore generally, $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$ for an odd prime $p$. Gauss first proved this.\n\n## Explanations\n\n### L1 human\n\nFor a prime $p$, one residue modulo $p$ has powers that give all the nonzero residues.\nSuch a residue is a primitive root.\nSo the multiplicative group modulo $p$ is cyclic.\n",
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