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  "body": "\n## Proof\n\n1. Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a subgroup of $G$.\n2. For $a, b \\in G$, let $a \\sim b$ when $a^{-1}b \\in H$. Since $H$ contains the identity, inverses and products, this relation is reflexive, symmetric and transitive.\n3. The class of $a$ under $\\sim$ is the [[left coset|03DEH43F4H]] $aH$, because $a^{-1}b \\in H$ exactly when $b \\in aH$.\n4. The [[equivalence classes form a partition|01HT2CH8BH]] of $G$. Thus $G$ is the disjoint union of $k$ distinct left cosets for some integer $k \\ge 1$.\n5. [[Each left coset has $|H|$ elements|0DKMBWTD7P]].\n6. Thus $|G| = k\\,|H|$, and $|H|$ divides $|G|$.\n\n## Idea\n\nThe left cosets of the subgroup cut the group into blocks of equal size. The order of the group is the number of blocks times the size of one block.\n",
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  "name": "Proof of Lagrange's theorem",
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