Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 19I / a

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- Solution a

Solution

Let

χ_{reg}

be the character of the regular representation. It has

χ_{reg} (1) = ∣ G ∣

and vanishes away from the identity. The multiplicity of an irreducible character

χ_{j}

in it is

⟨ χ_{reg}, χ_{j} ⟩ = \frac{1}{∣ G ∣} ∣ G ∣ χ_{j} (1) = n_{j} .

Thus the regular representation decomposes as

⨁_{j} n_{j} ρ_{j}

. Comparing dimensions gives

∣ G ∣ = j = 1 \sum k n_{j}^{2} .

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