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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen $|G| = [G : H] \\cdot |H|$, where $[G : H]$ is the [[index|0DTXBQPA2S]] of $H$ in $G$.\n\n## Remarks\n\nThe left cosets of $H$ partition $G$, and [[each coset has $|H|$ elements|0DKMBWTD7P]].\nThis statement gives [[Lagrange's theorem|3C53TD]]: the order of $H$ divides the order of $G$.\nThe formula $|G| = [G : H] \\cdot |H|$ is also true for infinite groups as an equation of cardinal numbers.\n\n## Explanations\n\n### L1 human\n\nCut a finite group into the left cosets of a subgroup.\nEach piece has the same size as the subgroup, and the number of pieces is the index.\nSo the size of the group is the number of pieces times the size of one piece.\n",
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