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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

1 Answer 1

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".)

What you are trying to prove follows almost immediately from this description of the induced module. Indeed, in the aforementioned lecture notes, your exercise appears as Example 1 on the following page (with no further explanation). If you can't see how to deduce this, you should probably book up (and possibly ask further questions) on the definition of an induced module.

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