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

Statement

Suppose $H$ is a normal subgroup of a group $G$. Denote by $\psi$ the trivial representation of $H$. Then, the induced representation $\operatorname {Ind} _{H}^{G}\psi$ can be viewed as a permutation representation. Specifically, the permutation representation is the group action on the left coset space $G/H$ by left multiplication.

Related facts

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