= Solution
Let $\zeta=e^{2\pi i/n}$. The <group homomorphism> $\mathbb Z_n\to U(1)$ given by $[k]\mapsto\zeta^k$ is injective, identifying the <finite cyclic group> with the <subgroup> of $n$th <roots of unity> in the <circle group>.
The continuous <irreducible representations> of the <circle group> are the <one-dimensional representations> $R_Q(e^{i\theta})=e^{iQ\theta}$ with $Q\in\mathbb Z$. One way to see this is to make the <representation> <unitary> by averaging, simultaneously diagonalize its commuting <unitary matrices>, and impose $R_Q(e^{i(\theta+2\pi)})=R_Q(e^{i\theta})$. Its <restriction of a representation> is
$$
R_Q(\zeta^k)=e^{2\pi iQk/n}=e^{2\pi iqk/n},\qquad
\boxed{q=Q\pmod n.}
$$
Thus each <irreducible representation> remains one-dimensional, and <representations> whose integer labels differ by a multiple of $n$ become equivalent. Every <irreducible representation> of the <finite cyclic group> occurs this way.
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