Brauer's characterization of characters ($f\text{ virtual on }G⟺f_E\text{ virtual on every elementary }E\le G$)

A complex class function $f$ on a finite group is a generalized character if and only if $f_E$ is a generalized character of every elementary subgroup. Equivalently, $⟨f_E,\alpha {⟩}_E\in \mathbb{Z}$ for every such $E$ and $\alpha \in \mathrm{Irr}(E)$. Ordinary characters additionally have nonnegative global irreducible multiplicities; integrality of the virtual-character lattice is the conclusion of this criterion.

Suppose every nonidentity element of a finite group $K$ is either a pi-element or an element for complementary primes. An elementary subgroup cannot have nontrivial factors of both kinds, since the product of nonidentity commuting coprime-order elements would have mixed prime order. Choose an integer $d$ congruent to $1$ modulo $|K{|}_{\pi }$ and to $0$ modulo $|K{|}_{{\pi }^{\prime }}$. Set $f(1)=d$, $f=1$ on nonidentity pi-elements and $f=0$ on the other nonidentity elements. On each elementary subgroup this is an integer combination of the trivial and regular characters. Brauer's characterization of characters supplies the asserted generalized character.