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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]].\nThen every complex [[character|RMWMYBY8TC]] of $G$ is a rational linear combination of [[induced|JJP4JFH3C6]] characters $\\operatorname{Ind}_C^G \\theta$.\nHere $C$ runs over the [[cyclic|0KVGH0J5B3]] subgroups of $G$ and $\\theta$ over the characters of $C$.\n\n## Remarks\n\nFor the proof, let $\\psi$ be a class function on $G$ orthogonal to all characters induced from cyclic subgroups.\nBy [[Frobenius reciprocity|PAP76H0ND9]], the restriction of $\\psi$ to each cyclic subgroup is orthogonal to all its characters, so it is zero.\nEvery element of $G$ lies in a cyclic subgroup, so $\\psi = 0$.\nHence the induced characters span the space of class functions over $\\C$, and a rationality argument gives rational coefficients.\nRational coefficients are needed in general.\nFor example, let $V = \\Z/2 \\times \\Z/2$ be the Klein four-group.\nIts cyclic subgroups are proper, so every character induced from one of them has even degree, and so has every integer combination of such characters.\nHence the trivial character $1_V$, of degree $1$, is not an integer combination.\nBut $1_V = \\frac{1}{2}\\sum_{C} \\operatorname{Ind}_C^V 1_C - \\frac{1}{2}\\operatorname{Ind}_{1}^{V} 1$, with $C$ running over the three subgroups of order $2$.\n[[Brauer's theorem on induced characters|V8JVWE87TE]] reaches integer coefficients by using elementary subgroups instead of cyclic ones.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that all characters of a finite group come from characters of its cyclic subgroups, the simplest subgroups there are, by induction and by taking rational combinations.\nSo, up to fractions, cyclic subgroups control the whole character theory.\n\n### L2 human\n\nArtin used this to show that a power of every Artin L-function is a quotient of products of abelian L-functions, so that some power of it is meromorphic.\nBrauer's theorem later removed the power.\nThe theorem is a statement about the rational character ring: tensored with $\\Q$, the induction maps from cyclic subgroups are jointly surjective.\n",
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