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  "body": "\n## Statement\n\nLet $G$ be a finite [[abelian group|082GHNPJ53]].\nThen $G$ is [[cyclic|0KVGH0J5B3]] if and only if the [[exponent|0ZDW0H0VX4]] of $G$ equals $|G|$.\n\n## Remarks\n\nIf $G$ is cyclic with generator $g$, then the order of $g$ is $|G|$, so the exponent is $|G|$.\nFor the converse, write $G \\cong \\mathbb{Z}/k_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/k_u\\mathbb{Z}$ with $1 \u003c k_1 \\mid k_2 \\mid \\cdots \\mid k_u$, by the [[classification|07CQR9BW3R]]. The exponent is $k_u$ and the order is $k_1 \\cdots k_u$, so they are equal only when $u \\leq 1$.\nThe abelian condition is necessary: $S_3$ has exponent $6$ and order $6$, but it is not cyclic.\nFor example, $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is cyclic if and only if $\\lambda(n) = \\varphi(n)$.\n\n## Explanations\n\n### L1 human\n\nIn a cyclic group, one element has order equal to the size of the group.\nFor a finite abelian group, it is enough that the exponent equals the size.\nThen the group is cyclic.\n",
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