 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ia/paper-3/5d/i/solution

g^{3}

and the six reflections

g^{j} h

. Thus the seven order-two subgroups are

⟨ g^{3} ⟩, ⟨ g^{j} h ⟩ (j = 0, 1, 2, 3, 4, 5) .

The half-turn is central, so only $⟨ g^{3} ⟩$ is normal. Indeed

g (g^{j} h) g^{- 1} = g^{j + 2} h

, a different reflection; an order-two subgroup has only one nonidentity element, so each reflection subgroup fails the normal subgroup condition.

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