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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $g \\in G$.\nThen $g$ and its inverse $g^{-1}$ have the same [[order|0WE2NHZJ7B]].\nIn particular, $g$ has infinite order if and only if $g^{-1}$ has infinite order.\n\n## Remarks\n\nFor each positive integer $n$, $(g^{-1})^n = (g^n)^{-1}$, and $g^n = e$ if and only if $(g^n)^{-1} = e$.\nFor example, in $\\{1, i, -1, -i\\}$, both $i$ and $i^{-1} = -i$ have order $4$.\n\n## Explanations\n\n### L1 human\n\nThe inverse of an element walks the same path back to the identity, in the opposite direction.\nIt needs the same number of steps.\nA rotation by $90$ degrees and a rotation by $-90$ degrees both have order $4$.\n",
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