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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]].\nLet $p$ be a prime.\nA subgroup $E$ of $G$ is $p$-elementary if it is a [[direct product|00HF4RJ5QN]] $E = C \\times P$ of two subgroups.\nHere $C$ is a [[cyclic group|0KVGH0J5B3]] of order prime to $p$ and $P$ is a [[$p$-group|06QYRTX9KQ]].\nA subgroup is elementary if it is $p$-elementary for some prime $p$.\nThen every complex [[character|RMWMYBY8TC]] of $G$ is an integer linear combination of [[induced|JJP4JFH3C6]] characters $\\operatorname{Ind}_E^G \\lambda$.\nHere $E$ runs over the elementary subgroups of $G$ and $\\lambda$ over the degree $1$ characters of $E$.\n\n## Remarks\n\nThe proof shows that the trivial character $1_G$ lies in the subgroup $I$ of the ring of virtual characters spanned by characters induced from elementary subgroups.\nSince $I$ is an ideal, by the formula $\\operatorname{Ind}(\\theta) \\cdot \\psi = \\operatorname{Ind}(\\theta \\cdot \\psi|_E)$, it then contains every character.\nThe passage from all characters of elementary subgroups to degree $1$ characters uses that elementary groups are nilpotent, so their irreducible characters are induced from degree $1$ characters.\nThe coefficients cannot in general be taken nonnegative.\nIf $G$ is not elementary, every elementary subgroup $E$ is proper, so each $\\operatorname{Ind}_E^G \\lambda$ has degree $[G : E] \\ge 2$.\nThen no combination with nonnegative integer coefficients equals the trivial character, of degree $1$.\nCompared with [[Artin's induction theorem|2CABBY4EEB]], which uses cyclic subgroups and rational coefficients, Brauer's theorem uses larger subgroups and gains integer coefficients.\n\n## Explanations\n\n### L1 human\n\nEvery character of a finite group can be assembled from very simple pieces: one-dimensional characters of small, almost abelian subgroups, pushed up to the whole group by induction.\nThe price is that one must allow subtraction as well as addition.\n\n### L2 human\n\nA main use is in number theory: Brauer deduced that Artin L-functions extend to meromorphic functions on the whole complex plane, since this holds for Hecke L-functions of one-dimensional characters.\nThe theorem also gives Brauer's characterisation of characters: a class function is a virtual character if and only if its restriction to every elementary subgroup is a virtual character.\nFrom this one obtains that every complex representation of $G$ can be realised over the field $\\Q(\\zeta_m)$, where $m$ is the exponent of $G$.\n",
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