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  "body": "\n## Summary\n\nThe transfer is a [[group homomorphism|0JYTG5K9Y4]] from a group $G$ to the [[abelianisation|0AQZV0905N]] of a subgroup $H$ of finite [[index|0DTXBQPA2S]].\nIt is built from the way $G$ permutes the [[left cosets|03DEH43F4H]] of $H$.\nThe transfer gives normal subgroups and abelian quotients of $G$ from information about $H$, usually a [[Sylow subgroup|0053CQE9Y2]].\nIt proves results such as Burnside's normal $p$-complement theorem and the [[focal subgroup theorem|H1BSG0J5G1]].\n\n## Explanations\n\n### L1 human\n\nLet $H$ be a subgroup of $G$ with $n$ cosets.\nAn element $g$ of $G$ shuffles the cosets, and each step of the shuffle leaves a small correction term in $H$.\nThe transfer multiplies these $n$ correction terms together, in a version of $H$ where the order of multiplication does not matter.\nWhen $G$ is abelian, the result is simply $g^n$.\nFor the integers with $H$ the multiples of $n$, the transfer sends $1$ to $n$.\n",
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