Unitary dual of $\{0\}$ and $\mathbb R$

Question

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively.

Thank you in advance

Answer 1

For a finite group $G$, the number of inequivalent irreducible unitary representations equals the number of conjugacy classes of $G$. The trivial group only has one conjugacy class. The second question on $\mathbb{R}$ has been answered at this MSE question.