 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-2/19g/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The irreducible characters are an orthonormal basis of the space of complex class functions for the inner product

⟨ f, h ⟩ = \frac{1}{∣ G ∣} \sum_{g \in G} \overline{f (g)} h (g) .

Thus the row orthogonality relation and completeness statement are

\frac{1}{∣ G ∣} g \in G \sum \overline{χ_{i} (g)} χ_{k} (g) = δ_{ik}, f = i \sum ⟨ χ_{i}, f ⟩ χ_{i} .

In particular the number of irreducible characters equals the number of conjugacy classes. In class-representative notation, the orthogonality relation is

\sum_{j} \overline{χ_{i} (g_{j})} χ_{k} (g_{j}) /∣ Z (g_{j}) ∣ = δ_{ik}

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