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 representationExplicitly $W\mapsto 1$, $X\mapsto i$, $\hat{W}\mapsto -1$ and $\hat{X}\mapsto -i$. 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.