Solution
= Solution
For $G_1=\langle X\rangle$, define the one-dimensional complex <group representation>
$$
\boxed{\rho(X^j)=i^j,\qquad j=0,1,2,3.}
$$
Explicitly $W\mapsto1$, $X\mapsto i$, $\widehat W\mapsto-1$ and $\widehat X\mapsto-i$. 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>.