Brauer's theorem on induced characters

Remarks

The 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. Since $I$ is an ideal, by the formula $\operatorname{Ind}(\theta) \cdot \psi = \operatorname{Ind}(\theta \cdot \psi|_E)$, it then contains every character. The 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. The coefficients cannot in general be taken nonnegative. If $G$ is not elementary, every elementary subgroup $E$ is proper, so each $\operatorname{Ind}_E^G \lambda$ has degree $[G : E] \ge 2$. Then no combination with nonnegative integer coefficients equals the trivial character, of degree $1$. Compared with Artin's induction theorem, which uses cyclic subgroups and rational coefficients, Brauer's theorem uses larger subgroups and gains integer coefficients.

Going deeper

For a student

A 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. The 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. From this one obtains that every complex representation of $G$ can be realised over the field $\Q(\zeta_m)$, where $m$ is the exponent of $G$.