 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/19i/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

\sum_{i = 1}^{r} a_{i}^{2} = ⟨ χ ↓_{H}, χ ↓_{H} ⟩_{H} = \frac{1}{∣ H ∣} \sum_{h \in H} ∣ χ (h) ∣^{2} .

Since

χ

G

is one, and hence

\sum_{i} a_{i}^{2} \leq \frac{1}{∣ H ∣} \sum_{g \in G} ∣ χ (g) ∣^{2} = \frac{∣ G ∣}{∣ H ∣} = [G : H] .

The omitted summands are nonnegative, so the character restriction norm bound is an equality precisely when

χ (g) = 0 for every g \in G ∖ H .

 New to topics? Read the docs here!