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  "body": "\n## Statement\n\nLet $n$ be a positive integer and let $G$ be a [[cyclic group|0KVGH0J5B3]] of order $n$.\nFor each integer $a$ with $\\gcd(a, n) = 1$, the map $\\sigma_a \\colon x \\mapsto x^a$ is an automorphism of $G$.\nThe map $a \\bmod n \\mapsto \\sigma_a$ is an [[isomorphism|08MG7H8JW5]] from the [[group of units|0FJ90Y9926]] $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ to $\\operatorname{Aut}(G)$.\n\n## Remarks\n\nAn automorphism of $G = \\langle g \\rangle$ is determined by the image of $g$, and this image must be a generator $g^a$ with $\\gcd(a, n) = 1$.\nThus $|\\operatorname{Aut}(G)| = \\varphi(n)$, where $\\varphi$ is Euler's totient function.\nFor the infinite cyclic group, $\\operatorname{Aut}(\\mathbb{Z}) \\cong \\{1, -1\\}$.\n\n## Explanations\n\n### L1 human\n\nAn automorphism of a cyclic group must send a generator to a generator.\nEach choice of a generator gives exactly one automorphism.\nFor $\\mathbb{Z}/5\\mathbb{Z}$, the generators are $1, 2, 3, 4$, so there are four automorphisms, given by multiplication by $1, 2, 3, 4$.\n",
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