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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe exponent of $G$ is the least positive integer $n$ such that $g^n = e$ for each $g \\in G$.\nIf no such integer exists, then $G$ has infinite exponent.\n\n## Remarks\n\nWhen the exponent is finite, it is the least common multiple of the [[orders of the elements|0WE2NHZJ7B]] of $G$.\nThe exponent of a finite group divides the order of the group, by [[Lagrange's theorem|3C53TD]].\nThe exponent of the cyclic group of order $n$ is $n$, and the exponent of the [[Klein four-group|0PMRQJVTME]] is $2$.\nThe exponent of $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is the Carmichael function $\\lambda(n)$.\n\n## Explanations\n\n### L1 human\n\nThe exponent is the smallest power that sends every element to the identity.\nIt is a common multiple of all the element orders.\nA finite group always has a finite exponent, and the exponent divides the order.\n",
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