Faithful representations and tensor products

Let G be a finite group (or a compact Lie group). Prove that if \rho:G\to\mathrm{GL}(V) is a faithful finite dimensional complex representation of G then any irreducible representation embeds in some tensor product of V.

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Faithful representations and tensor products

Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group D_{2n} with 2n elements over \mathbb{C}. Classify all irreducible representations for D_{2n} over \mathbb{Q}.

Be rational