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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $g \\in G$ be an element of finite [[order|0WE2NHZJ7B]] $n$.\nLet $k$ be an integer.\nThen the order of $g^k$ is $n / \\gcd(n, k)$.\n\n## Remarks\n\nHere $\\gcd(n, 0) = n$, so that $g^0 = e$ has order $1$.\nIn particular, $g^k$ has order $n$ if and only if $\\gcd(n, k) = 1$.\nFor example, if $g$ has order $12$, then $g^8$ has order $12 / 4 = 3$.\n\n## Explanations\n\n### L1 human\n\nA power of an element returns to the identity faster than the element itself, or at the same time.\nIf $g$ has order $12$, then $g^8$ is at the identity after $k$ steps exactly when $12$ divides $8k$, that is, when $3$ divides $k$.\nThus $g^8$ has order $3$.\n",
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