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I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of:

Let $G$ be a finite group and let $R[G]$ be the representation ring of $G$: i.e. the free abelian group on the irreducible representations of $G$, with multiplication given by tensor product of the irreducible representations. Let $I$ be the subset of $R[G]$ consisting of the $\mathbb{Z}$-span of representations induced from characters on elementary subgroups. Then $I$ is an ideal of $R[G]$.

Why is $I$ an ideal?

In particular, if $a\in R[G]$ and $b\in I$, why is $ab \in I$? And, does this depend on the particular choice of generators of $I$ as a $\mathbb{Z}$-module or would it be true for the $\mathbb{Z}$-span of any set from $R[G]$?

Thanks in advance.

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To prove that it is enough that you show that given any rep. $V$ and a rep $W$ induced from an elementary subgroup, the product $V\otimes W$ is induced from an elementary subgroup. The statement thus reduces to a very concrete fact about representations—if you started with an arbitrary ideal, no such thing would be possible! –  Mariano Suárez-Alvarez Apr 30 '12 at 19:27

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