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  "body": "\n## Statement\n\nLet $G$ be a [[Frobenius group|JXS91FGFHY]] acting on a finite set $\\Omega$, and let $H = G_\\omega$ be the [[stabiliser|0AYMP41THN]] of a point $\\omega \\in \\Omega$.\nLet $K$ be the set of elements of $G$ that fix no point of $\\Omega$, together with the identity $e$.\nThen $K$ is a [[normal subgroup|0CB0DX5BVV]] of $G$, $K$ acts regularly on $\\Omega$, and $G$ is the [[semidirect product|0H3WZXCZNF]] $K \\rtimes H$.\n\n## Remarks\n\nCounting is easy.\nLet $n = \\abs{\\Omega}$; the $n$ point stabilisers are the [[conjugates|0H11ZPX4ND]] of $H$, and any two of them meet only in $e$.\nSo their union has $n(\\abs{H} - 1) + 1$ elements, and $K$ has $\\abs{G} - n\\abs{H} + n = n$ elements.\nThe hard part is that $K$ is closed under multiplication.\nThe standard proof uses characters.\nEach irreducible character of $H$ is extended to a character of $G$, and $K$ is the intersection of the kernels of these characters.\n\nExample: for the maps $x \\mapsto ax + b$ with $a \\ne 0$ acting on $\\F_q$, the kernel $K$ is the group of translations $x \\mapsto x + b$, and the complement $H$ is the group of maps $x \\mapsto ax$.\nFor $S_3$ acting on $3$ points, $K = A_3$.\n\nJohn Thompson later proved that the Frobenius kernel is always [[nilpotent|0Q7MMQHC1P]].\n\n## Explanations\n\n### L1 human\n\nIn a Frobenius group, the elements that move every point look like a loose collection.\nFrobenius's theorem says they form a subgroup, and in fact a normal one.\nSo every Frobenius group splits into two pieces: the kernel, which moves points around freely, and a point stabiliser.\n\n### L2 human\n\nThe theorem is a famous example of the power of character theory: its statement is purely about permutations, but the standard proof needs characters.\nIt gives the structure $G = K \\rtimes H$ with $\\abs{K} = \\abs{\\Omega}$ and $\\abs{H}$ dividing $\\abs{K} - 1$, since $H$ acts on $K \\setminus \\set{e}$ by conjugation with all orbits of size $\\abs{H}$.\nThe Frobenius kernel and complement are central in the study of finite groups, for example in the proof of the Feit–Thompson theorem.\n",
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