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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $g \\in G$.\nThe cyclic subgroup generated by $g$ is the set $\\langle g \\rangle = \\{g^k : k \\in \\mathbb{Z}\\}$ of all integer powers of $g$.\n\n## Remarks\n\nBy [[the laws of exponents|00DRK90Z6K]], $g^j (g^k)^{-1} = g^{j-k}$, thus $\\langle g \\rangle$ is a [[subgroup|0X1JXT47GA]] of $G$.\nIt is the [[subgroup generated by|0RP7VWQ0YN]] the set $\\{g\\}$.\nIn additive notation, $\\langle g \\rangle = \\{kg : k \\in \\mathbb{Z}\\}$, for example $\\langle 3 \\rangle = 3\\mathbb{Z}$ in $(\\mathbb{Z}, +)$.\n\n## Explanations\n\n### L1 human\n\nTake one element and all its powers, positive and negative, and the identity.\nThese powers form a subgroup, the smallest one that contains the element.\nIn the integers with addition, the cyclic subgroup generated by $3$ is the set of multiples of $3$.\n",
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