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  "body": "\n## Statement\n\nLet $G$ and $H$ be [[cyclic groups|0KVGH0J5B3]] of the same [[order|0B85S2V2DR]], finite or infinite.\nThen $G$ and $H$ are isomorphic: there is a bijection $\\varphi \\colon G \\to H$ with $\\varphi(ab) = \\varphi(a) \\varphi(b)$ for all $a, b \\in G$.\n\n## Remarks\n\nIf $g$ generates $G$ and $h$ generates $H$, an isomorphism is $\\varphi(g^k) = h^k$ for $k \\in \\mathbb{Z}$.\nThis map is well defined because $g^a = g^b$ exactly when $h^a = h^b$, by [[the order of a cyclic subgroup|031M8HX7DA]].\nThus each cyclic group of order $n$ is isomorphic to $\\mathbb{Z}/n\\mathbb{Z}$, and each infinite cyclic group is isomorphic to $\\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nUp to the names of the elements, there is only one cyclic group of each size.\nTo match two of them, send the generator of one to the generator of the other, and send each power to the same power.\nFor example, the rotations of a hexagon and the integers modulo $6$ are the same group in two costumes.\n",
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