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  "body": "\n## Statement\n\nLet $G$ be a [[cyclic group|0KVGH0J5B3]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen $H$ is cyclic.\n\n## Remarks\n\nLet $g$ generate $G$.\nIf $H \\neq \\{e\\}$, let $m$ be the smallest positive integer with $g^m \\in H$; then division with remainder shows that $H = \\langle g^m \\rangle$.\nFor example, each subgroup of $(\\mathbb{Z}, +)$ has the form $m\\mathbb{Z}$, generated by $m$.\n\n## Explanations\n\n### L1 human\n\nA subgroup of a cyclic group is generated by the smallest positive power of the generator that it contains.\nAny other power in the subgroup is a multiple of this one, or the subgroup would contain a smaller positive power.\nThus the subgroup is again cyclic.\n",
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  "name": "Subgroups of cyclic groups are cyclic",
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