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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \\in G$.\nThen the map $h \\mapsto gh$ is a bijection from $H$ to the [[left coset|03DEH43F4H]] $gH$. The map $h \\mapsto hg$ is a bijection from $H$ to the [[right coset|0NE1DK7H4T]] $Hg$.\nIn particular, $|gH| = |Hg| = |H|$.\n\n## Remarks\n\nThe inverse of $h \\mapsto gh$ is $x \\mapsto g^{-1}x$.\nThis fact and the partition of $G$ into left cosets give the proof of [[Lagrange's theorem|3C53TD]].\n\n## Explanations\n\n### L1 human\n\nMultiplying by a fixed element never makes two different elements equal, by cancellation.\nThus a coset has exactly as many elements as the subgroup.\nAll cosets of a subgroup have the same size.\n",
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