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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a [[prime number|01826KMG2H]], and let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$.\nLet $G'$ be the [[commutator subgroup|0ASGSTXSBW]] of $G$.\nThe *focal subgroup* of $P$ in $G$ is [[generated|0RP7VWQ0YN]] by all $x^{-1} y$ with $x, y \\in P$ and $y$ [[conjugate|0DXD4QSZ8A]] to $x$ in $G$:\n$$\\operatorname{Foc}_G(P) = \\langle x^{-1} y : x, y \\in P, \\ y = g^{-1} x g \\text{ for some } g \\in G \\rangle.$$\nThen\n$$P \\cap G' = \\operatorname{Foc}_G(P).$$\n\n## Remarks\n\nThe inclusion $\\operatorname{Foc}_G(P) \\subseteq P \\cap G'$ is easy, because $x^{-1} g^{-1} x g$ is a commutator.\nThe other inclusion uses the [[transfer|3R2Y906QAQ]] $V \\colon G \\to P/P'$ and a formula for $V(x)$ with $x \\in P$ in terms of conjugates of powers of $x$ that lie in $P$.\n\nExample: $G = S_3$.\nFor $p = 3$, $P = A_3$, and $(1\\,2\\,3)$ is conjugate to $(1\\,3\\,2)$, so $\\operatorname{Foc}_G(P)$ contains $(1\\,2\\,3)^{-1}(1\\,3\\,2) = (1\\,2\\,3)$ and equals $A_3 = P \\cap G'$.\nFor $p = 2$, $P = \\set{e, (1\\,2)}$, the only element of $P$ conjugate to $(1\\,2)$ is $(1\\,2)$ itself, so $\\operatorname{Foc}_G(P) = \\set{e}$; indeed $P \\cap A_3 = \\set{e}$.\n\nAs a consequence, the largest abelian $p$-group that is a quotient of $G$ is isomorphic to $P / \\operatorname{Foc}_G(P)$.\nSo the abelian $p$-quotients of $G$ can be read off from $P$ and the way $G$ conjugates the elements of $P$.\n\n## Explanations\n\n### L1 human\n\nThe commutator subgroup measures how far a group is from being abelian.\nThe theorem says that its part inside a Sylow subgroup $P$ is generated by the \"jumps\" between elements of $P$ that the whole group makes conjugate.\nSo a question about the whole group reduces to a question about $P$ and conjugation.\n\n### L2 human\n\nThe focal subgroup theorem is a central result of transfer theory.\nIt tells when a finite group has a nontrivial abelian $p$-group as a quotient: exactly when $\\operatorname{Foc}_G(P) \\ne P$.\nLike Burnside's normal $p$-complement theorem, it is proved with the transfer, and it leads to the study of fusion, that is, of which elements of $P$ are conjugate in $G$.\nThe [[Z* theorem|2XCJ8NNMXP]] is another result about fusion, for involutions.\n",
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