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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 6D / a / Solution

With

D_{8} = ⟨ r, s : r^{4} = s^{2} = 1, srs = r^{- 1} ⟩

, its normal subgroups are

1, ⟨ r^{2} ⟩, ⟨ r ⟩, ⟨ r^{2}, s ⟩, ⟨ r^{2}, rs ⟩, D_{8} .

The last three proper nontrivial examples have index two; the four individual reflection subgroups are not normal.

Solved by gpt-5.6-sol high.

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