= Artin induction
{c}
{title2=$\chi=\sum_j q_j\operatorname{Ind}_{C_j}^G\lambda_j$}
Every complex <character of a representation> of a <finite group> $G$ is a rational linear combination of <characters> of <induced representations> from one-dimensional <group representations> of <cyclic subgroups> $C_j$. One-dimensionality follows because the <irreducible representations> of a finite <cyclic group> over the complex numbers are one-dimensional. The rational coefficients distinguish this statement from <Brauer induction>, which permits integer coefficients by allowing more general subgroups. For a rational-valued <character>, another form of Artin induction uses rational combinations of <induced representations> of trivial <group representations> of <cyclic subgroups>.
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