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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $H$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $H$.\nThen $G = N_G(P) H$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$ in $G$.\n\n## Remarks\n\nFor $g \\in G$, the subgroup $gPg^{-1}$ lies in $H$, because $H$ is normal, and it is a Sylow $p$-subgroup of $H$.\nBy the [[second Sylow theorem|09GCQ5STVJ]] in $H$, there is $h \\in H$ with $gPg^{-1} = hPh^{-1}$, so $h^{-1}g \\in N_G(P)$.\nThe argument is used in the proof that a finite [[nilpotent group|0Q7MMQHC1P]] is the [[direct product of its Sylow subgroups|00Y9N80VSS]].\n\n## History\n\nGiovanni Frattini used the argument in a paper from 1885.\nHe took it, as he said himself, from a paper of Alfredo Capelli from 1884.\n\n## Explanations\n\n### L1 human\n\nConjugation by an element of $G$ moves $P$ to another Sylow subgroup of $H$.\nAn element of $H$ can move it back.\nSo each element of $G$ is an element of $H$ times an element that normalises $P$.\n",
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