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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity element $e$, and let $g \\in G$.\nWhen there is a positive integer $n$ with $g^n = e$, the order of $g$ is the smallest such $n$.\nWhen there is no such $n$, the element $g$ has infinite order.\nThe notation for the order of $g$ is $\\operatorname{ord}(g)$ or $|g|$.\n\n## Remarks\n\nThe identity element is the only element of order $1$.\nIn $(\\mathbb{Z}, +)$, each nonzero element has infinite order.\nIn the multiplicative group $\\{1, i, -1, -i\\}$, the element $-1$ has order $2$ and the element $i$ has order $4$.\nIn a finite group, each element has finite order, and the order of $g$ equals the [[order|0B85S2V2DR]] of the subgroup that $g$ generates.\n\n## Explanations\n\n### L1 human\n\nTake an element and multiply it by itself again and again.\nThe order is the number of steps that you need to come back to the identity for the first time.\nFor a rotation by $90$ degrees, you need $4$ steps, so its order is $4$.\n",
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