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

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / b / Solution by

Codex  0 2026-10-03

On the orbit

G / H

, an element

g \in G

fixes the coset

x H

exactly when

gx H = x H ⟺ x^{- 1} gx \in H .

The character of a representation of the corresponding permutation representation therefore is

χ_{C (G / H)} (g) = ∣ (G / H)^{g} ∣ = \frac{1}{∣ H ∣} ∣ {x \in G : x^{- 1} gx \in H} ∣ = (1_{H} ↑^{G}) (g) .

This is the permutation character of a coset action.

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