# Induced representation

Let H be a subgroup of G and V the trivial representation of C[H]. Prove that we have an isomorphism of C[G]-modules between $\mathrm{Ind}_{H}^{G}(V)$ and C[G/H].

For instance, one of the standard descriptions of the induced module $\operatorname{Ind}_H^G(V)$ is as a direct sum $\bigoplus_{s \in S} s V$ of copies of $V$ indexed by a set $S$ of coset representatives of $G/H$ with a certain natural $G$-action. See for instance Lemma 2.1 in these notes. (I have no particular connection to these course notes; they just came up as one of the first hits upon googling "induced module".)