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  "body": "\n## Definition\n\nLet a finite group $G$ act [[transitively|02GK6HZYBT]] on a finite set $\\Omega$ with $\\abs{\\Omega} \\ge 2$.\n$G$ is a *Frobenius group* (for this action) if:\n\n1. no element $g \\ne e$ of $G$ fixes two distinct points of $\\Omega$, and\n2. some element $g \\ne e$ of $G$ fixes a point of $\\Omega$.\n\n## Remarks\n\nEquivalently, a finite group $G$ is a Frobenius group if it has a [[subgroup|0X1JXT47GA]] $H$ with $\\set{e} \\ne H \\ne G$ and $H \\cap gHg^{-1} = \\set{e}$ for all $g \\in G \\setminus H$.\nThen $H$ is a point stabiliser for the action on the left cosets of $H$, and it is called a *Frobenius complement*.\nThe set $K$ of elements that fix no point, together with $e$, is the *Frobenius kernel*; by [[Frobenius's theorem|XK7H3YXBQT]] it is a normal subgroup.\n\nExample: the group of maps $x \\mapsto ax + b$ with $a \\ne 0$, acting on a [[finite field|0ZXZ82FTX2]] $\\F_q$ with $q \\ge 3$.\nIf $a \\ne 1$, the equation $ax + b = x$ has exactly one solution, and if $a = 1$ and $b \\ne 0$ it has none, so clause 1 holds.\nThe map $x \\mapsto ax$ with $a \\ne 0, 1$ fixes $0$, so clause 2 holds.\nFor $q = 3$ this is $S_3$ acting on $3$ points.\nFor odd $n \\ge 3$, the [[dihedral group|0JZ27DMJ5D]] of order $2n$ acting on the corners of a regular $n$-gon is also a Frobenius group.\n\nNon-example: $S_4$ acting on $\\set{1, 2, 3, 4}$.\nThe transposition $(1\\,2)$ fixes the two points $3$ and $4$, so clause 1 fails.\nThe cyclic group generated by $(1\\,2\\,3)$ acting on $\\set{1, 2, 3}$ satisfies clause 1 but fails clause 2.\n",
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