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  "body": "\n## Summary\n\nInduction builds a [[representation|WYGRXCG2RN]] of a group $G$ from a representation of a [[subgroup|0X1JXT47GA]] $H$.\nThe induced representation is a sum of copies of the given space, one for each left coset of $H$, which $G$ permutes.\nInduction is adjoint to restriction, by [[Frobenius reciprocity|PAP76H0ND9]].\nMany representations of finite groups are constructed this way.\n\n## Explanations\n\n### L1 human\n\nSuppose you know how a subgroup $H$ acts on a space $W$.\nTake one copy of $W$ for each coset of $H$ in $G$, and let $G$ shuffle the copies while acting inside them through $H$.\nThe result is a representation of the whole group.\nFor instance, inducing the trivial representation of $H$ gives the permutation representation of $G$ on the cosets $G/H$.\nInducing from the trivial subgroup gives the regular representation, where $G$ acts on its own group algebra.\n",
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