How many irreducible representations does SO(2) have?

Question

SO(2) is a continuous abelian group describing proper rotations in 2d space. Because all elements commute, each conjugacy class contains only one element. Thus, there are uncountably many classes that can be parameterized by the angle $0\le\phi<2\pi$. According to the group theory, the number of classes is equal to the number of irreducible representations (irreps). Thus, there should be uncountably many irreps. However, the irreps are known. They can be parametrized by integer (or half-integer) $J$, which has the meaning of the angular momentum in quantum mechanics. There are countably many of them. What are the remaining uncountably many representations? Am I missing something in these considerations?

Answer 1

The statement “the number of classes is equal to the number of irreducible representations” holds for finite groups, but not in general. And $SO(2,\mathbb R)$ is infinite.