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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThe index of $H$ in $G$, written $[G : H]$ or $|G : H|$, is the cardinality of the set of [[left cosets|03DEH43F4H]] of $H$ in $G$.\n\n## Remarks\n\nThe index is also the number of right cosets, because $gH \\mapsto Hg^{-1}$ is a bijection between the two sets of cosets.\nIn $(\\mathbb{Z}, +)$, the subgroup $n\\mathbb{Z}$ has index $n$ for each positive integer $n$.\nA subgroup of a finite group has index $|G| / |H|$, but an infinite group can have a subgroup of finite index, for example $2\\mathbb{Z}$ in $\\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nThe index counts how many disjoint copies of the subgroup you need to cover the whole group.\nFor the even integers inside the integers, you need two copies: the even and the odd integers.\nThus the index of the even integers is $2$.\n",
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