If cyclic boundary conditions are imposed on a periodic 1D lattice by identifying the two ends with \(N\) cycles between the boundaries, the translation group becomes a cyclic group of order \(N\). Show that the irreps of this group are of the form

\[\Gamma^{(p)}(C)=e^{2 \pi i p / N}(p=1,2,3, \ldots, N),\]

where \(C\) denotes group elements. Hint: Elements of the cyclic group of order \(N\) are \(C_{1}=c, C_{2}=c^{2}, C_{3}=c^{3}, \ldots, C_{N}=c^{N}=1\), because of the closure condition \(C_{N}=1\), where 1 is the group identity. Thus the group is abelian and irreps are 1D and labeled by complex numbers.