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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/10/e/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 10 / e / Solution by

Codex  0 2026-10-18

For any cyclic subgroup

H = ⟨ s ⟩

of order

n \in {1, 2, 4}

ρ_{1} (s^{j}) = e^{2 πij / n} \in G L (1, C), ρ_{2} (s^{j}) = (cos (2 πj / n) sin (2 πj / n) - sin (2 πj / n) cos (2 πj / n)) \in G L (2, R) .

Both preserve products because scalar phases and planar rotations add their angles. Each sends the generator to an element of exact order

n

, so both have trivial kernel and are faithful representations.

For

⟨ U ⟩ ≅ C_{4}

, take

s = U

: the generator images are

i

and

R (π /2)

. For each order-two subgroup take its nonidentity element

s

; the images are

- 1

and

- I_{2}

. For the trivial subgroup the unique images are

1

and

I_{2}

, which are faithful on a one-element group.

The one-dimensional representation of

C_{4}

here must use complex scalars. There is no faithful one-dimensional real representation of $C_{4}$ , since a nonzero real scalar of finite order is

1

or

- 1

, and neither has order four. The two-dimensional representations above are real.

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