 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-1/6b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The relations

M^{2} = I, R^{n} = I, MRM = R^{- 1}

imply

RM = M R^{- 1}

. Moving every occurrence of

M

to the left and reducing exponents therefore puts every word into one of the normal forms

R^{j} or M R^{j}, 0 \leq j < n .

Explicitly, with

θ = 2 π / n

R^{j} = (cos (j θ) sin (j θ) - sin (j θ) cos (j θ)),

and

M R^{j} = (cos (j θ) - sin (j θ) - sin (j θ) - cos (j θ)) .

The

n

rotations are distinct. The

n

reflected matrices are also distinct, and no reflected matrix equals a rotation because their determinants are

- 1

and

1

, respectively. Hence there are exactly

2 n

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