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  "body": "\n## Statement\n\nLet $K$ be a field and let $H$ be a finite-dimensional [[Hopf algebra|7XFYP0N84H]] over $K$.\nLet $B \\subseteq H$ be a Hopf subalgebra, that is a subalgebra with $\\Delta(B) \\subseteq B \\otimes B$ and $S(B) \\subseteq B$.\nThen $H$ is a [[free|0DZ9DKAKZR]] left $B$-module and a free right $B$-module.\nIn particular, $\\dim_K B$ divides $\\dim_K H$.\n\n## Remarks\n\nThe proof views $H$ as a relative Hopf module: a right $B$-module under multiplication with the compatible right $H$-coaction $\\Delta$.\nOne shows that a direct sum of finitely many copies of such a module is free over $B$.\nA Krull–Schmidt argument for finite-dimensional modules then shows that $H$ itself is free.\nExample: for a finite group $G$ and a subgroup $L$, the group algebra $KG$ is a free right $KL$-module with basis a set of representatives of the left cosets $gL$.\nThen $\\dim KL = \\abs{L}$ divides $\\dim KG = \\abs{G}$, which is [[Lagrange's theorem|3C53TD]].\nThe divisibility statement follows from freeness: if $H \\cong B^n$ as $B$-modules, then $\\dim H = n \\dim B$.\n\n## Explanations\n\n### L1 human\n\nLagrange's theorem says that the size of a subgroup divides the size of a finite group.\nThe Nichols–Zoeller theorem is the version for finite-dimensional Hopf algebras: the dimension of a Hopf subalgebra divides the dimension of the whole algebra.\nThe stronger statement is that the big algebra is a free module over the small one, just as a group is a disjoint union of cosets of a subgroup.\n\n### L2 human\n\nThe theorem is a basic tool in the classification of finite-dimensional Hopf algebras of small dimension.\nFor example, the group-like elements of $H$ span a Hopf subalgebra $KG(H)$, so the order of the group of group-likes divides $\\dim H$.\nThe [[index|0DTXBQPA2S]] analogue is the integer $\\dim H / \\dim B$, the rank of $H$ as a free $B$-module.\n",
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