= Solution
Let $\operatorname{Irr}(G)$ denote the <irreducible characters> of finite-dimensional complex <group representations> of $G$. A <generalized character>, or <virtual character>, is an integral linear combination of these <irreducible characters>. A <class function> is a function constant on <conjugacy classes>. For <class functions> use the <character inner product>
$$
\langle f,h\rangle_G=\frac1{|G|}\sum_{g\in G}f(g)\overline{h(g)}.
$$
A <p-elementary group> is a <direct product of groups> $P\times C$, where $P$ is a <p-group> and $C$ is a <cyclic group> of order coprime to $p$. An <elementary subgroup> is a <subgroup> of this form for some prime $p$. Either factor may be trivial. This notion is different from an <elementary abelian group>.
<Brauer's characterization of characters> says that a complex <class function> $f$ on $G$ is a <generalized character> exactly when its restriction to every <elementary subgroup> is a <generalized character> there. Equivalently,
$$
\boxed{f\in\mathbb Z\operatorname{Irr}(G)\quad\Longleftrightarrow\quad
\langle f_E,\alpha\rangle_E\in\mathbb Z\ \text{for every elementary }E\leq G,\ \alpha\in\operatorname{Irr}(E)}.
$$
The <character orthogonality> relations make these inner products the coefficients of the restrictions in their irreducible bases. The theorem detects the integral <character> lattice. An ordinary <character of a representation> additionally requires every global irreducible coefficient $\langle f,\chi\rangle_G$ to be nonnegative; integrality alone characterizes virtual <characters>, not necessarily actual <representations>.
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