 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/10/d/solution

For

G_{1} = ⟨ X ⟩

, define the one-dimensional complex group representation

ρ (X^{j}) = i^{j}, j = 0, 1, 2, 3.

Explicitly

W \mapsto 1

X \mapsto i

W \mapsto - 1

and

X \mapsto - i

. Since

i^{4} = 1

, the map is well-defined on exponents modulo four and preserves products. Its four images are distinct, so it is also a faithful representation. The same construction applies to either of the other cyclic subgroups.

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