 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/19i/b/i/solution

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

1_{X} = \sum_{x \in X} x

is fixed by every element of

G

, so

C 1_{X} ≅ C_{G}

V = {\sum_{x \in X} a_{x} x : \sum_{x \in X} a_{x} = 0}

is also

G

-invariant, and every vector has a unique decomposition into a constant vector and an element of

V

. Hence

C X = C_{G} \oplus V .

Here

V

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