Induced representation from trivial representation on normal subgroup factors through regular representation of quotient group

Statement

Suppose $H$ is a normal subgroup of a group $G$. Denote by $\psi$ the trivial representation of $H$. Denote by $\rho_{G/H}$ the regular representation of $G/H$ and by $q:G \to G/H$ the quotient map. Then, the induced representation from $H$ to $G$ of $\psi$ factors via $q$ to $\rho_{G/H}$, i.e.,:

$\operatorname{Ind}_H^G \psi = \rho_{G/H} \circ q$

Related facts

• Induced representation from regular representation of subgroup is regular representation of group
• Induced representation from trivial representation of subgroup is permutation representation for action on coset space