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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]] $n$.\nLet $H'$ be the [[commutator subgroup|0ASGSTXSBW]] of $H$, so that $H/H'$ is the [[abelianisation|0AQZV0905N]] of $H$.\nChoose elements $t_1, \\dots, t_n$ of $G$ such that $G$ is the disjoint union of the [[left cosets|03DEH43F4H]] $t_1 H, \\dots, t_n H$.\nFor $g \\in G$ and each $i$, there are a unique index $\\sigma_g(i)$ and a unique $h_i(g) \\in H$ with $g t_i = t_{\\sigma_g(i)} h_i(g)$.\nThe *transfer* is the map $V \\colon G \\to H/H'$ given by\n$$V(g) = h_1(g) h_2(g) \\cdots h_n(g) \\, H'.$$\n\n## Remarks\n\nThe product is taken in the abelian group $H/H'$, so its order does not matter.\nOne proves that $V$ is a [[group homomorphism|0JYTG5K9Y4]] and that it does not depend on the choice of $t_1, \\dots, t_n$.\n\nExample: if $G$ is abelian, then $H' = \\set{e}$ and $h_i(g) = t_{\\sigma_g(i)}^{-1} g t_i$.\nSince $\\sigma_g$ permutes $\\set{1, \\dots, n}$, the factors $t_i$ cancel, and $V(g) = g^n$.\nFor $G = \\Z$ and $H = n\\Z$, the transfer is $V(k) = nk$.\n\nNon-example: take the product $h_1(g) \\cdots h_n(g)$ in $H$ itself, without passing to $H/H'$.\nIn general it depends on the order of the factors and on the choice of the $t_i$, so it does not give a well-defined homomorphism to $H$.\n\nA main application is Burnside's normal $p$-complement theorem.\nLet $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of a finite group $G$ that lies in the [[centre|0DZ93493VN]] of its [[normaliser|0YGN3VAA0D]].\nThen $G$ has a normal subgroup $K$ with $G = KP$ and $K \\cap P = \\set{e}$.\n",
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