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  "body": "\n## Description\n\nLet $n \\geq 1$ be an integer.\nThe group of units of $\\mathbb{Z}/n\\mathbb{Z}$ is the [[group of units|0FJ90Y9926]] $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ of the ring $\\mathbb{Z}/n\\mathbb{Z}$ under multiplication.\nIts elements are the residue classes $[a]$ with $\\gcd(a, n) = 1$, and its identity is $[1]$.\nIt is a finite [[abelian group|082GHNPJ53]] of order $\\varphi(n)$.\n\n## Remarks\n\nFor $n = 8$, the group is $\\{[1], [3], [5], [7]\\}$, and the square of each element is $[1]$, so it is isomorphic to the [[Klein four-group|0PMRQJVTME]].\nFor $n = 16$, the group is isomorphic to $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/4\\mathbb{Z}$, so it is not cyclic.\nThe group is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$, with $p$ an odd prime and $k \\geq 1$. A generator is called a primitive root modulo $n$.\nOther notations are $(\\mathbb{Z}/n\\mathbb{Z})^*$, $U(\\mathbb{Z}/n\\mathbb{Z})$ and $\\mathbb{Z}_n^*$.\n\n## Explanations\n\n### L1 human\n\nMultiplication modulo $n$ does not make all of $\\mathbb{Z}/n\\mathbb{Z}$ a group, because $[0]$ has no inverse.\nThe classes coprime to $n$ do form a group under multiplication.\nThis group is important in number theory and in cryptography.\n",
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