Induced representation from regular representation of subgroup is regular representation of group

Statement

The basic statement is as follows: if $H$ is a subgroup of a group $G$, then the induced representation of $G$ from the regular representation of $H$ equals the regular representation of $G$.

The statement can be made in terms of characters (as class functions, in the sense of induced class function), in terms of permutation representations, or in terms of linear representations.

Related facts

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