{
  "aliases": [
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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $K$ be a subgroup of $H$.\nThen $[G : K] = [G : H] \\cdot [H : K]$, where $[\\cdot : \\cdot]$ is the [[index|0DTXBQPA2S]].\n\n## Remarks\n\nThe equation holds as an equation of cardinal numbers; in particular $[G : K]$ is finite if and only if $[G : H]$ and $[H : K]$ are both finite.\nFor example, in $(\\mathbb{Z}, +)$ with $H = 2\\mathbb{Z}$ and $K = 6\\mathbb{Z}$, the indices are $[\\mathbb{Z} : 6\\mathbb{Z}] = 6 = 2 \\cdot 3$.\n\n## Explanations\n\n### L1 human\n\nEach coset of $H$ in $G$ splits into $[H : K]$ cosets of $K$.\nSo the number of cosets of $K$ is the number of cosets of $H$ times $[H : K]$.\nIt is like a box with several drawers, each with the same number of compartments.\n",
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  "name": "Tower law for indices",
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