= Brauer's characterization of characters
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{title2=$f\text{ virtual on }G\Longleftrightarrow f_E\text{ virtual on every elementary }E\leq G$}
= Brauer's characterisation of characters
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{synonym}
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, $\langle f_E,\alpha\rangle_E\in\mathbb Z$ for every such $E$ and $\alpha\in\operatorname{Irr}(E)$. Ordinary characters additionally have nonnegative global irreducible multiplicities; integrality of the virtual-character lattice is the conclusion of this criterion.
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