# 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].

Thanks for your help.

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@user4190: If this is homework, please consider using the (homework) tag. Also, it is impolite to phrase help requests in the imperative mode ("Prove", "Show", "Solve"). –  Arturo Magidin Nov 29 '10 at 6:04
This is a fairly standard homework problem. What is your precise definition of induced module? What have you tried already? E.g., showing that they have the same character? –  Pete L. Clark Nov 29 '10 at 6:07

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".)