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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 8D / c / Solution

The standard isomorphisms are

SO (2) ⟶ B, (cos θ sin θ - sin θ cos θ) ⟼ e^{i θ},

and

R / Z ⟶ B, t + Z ⟼ e^{2 πi t} .

For

z \in B

, let

R_{z}

be the corresponding rotation. The identity

J R_{z} J = R_{z^{- 1}}

shows that

(z, ϵ) ⟼ {R_{z}, R_{z} J, ϵ = 1, ϵ = - 1

respects the multiplication of the generalized dihedral group. The decomposition in part (b) makes it bijective. Hence

O (2) ≅ D (B) .

Solved by gpt-5.6-sol high.

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