Definition of infinite cyclic group I'm having some conceptual issues with the infinite cyclic group $C_\infty$. Finite groups $C_n$ have a clear representation as integers $0,1,\cdots,n-1$ under addition $\pmod{n}$, or as the rotation group of the $n$-gon for $n\geq 3$. The rotation group of a circle, which is what I interpreted $C_\infty$ to be, has uncountable order since any real angle $[0,2\pi)$ is valid. This would make it isomorphic to $[0,2\pi)$ under addition $\pmod{2\pi}$. But online it says $(\mathbb{Z},+)$ is also isomorphic, which doesn't make sense to me because it has order $\aleph_0$. Also, the first group has two inverses $0$ and $\pi$, while this group only has $0$.
I'm guessing my interpretation is wrong. The textbook never defines what $C_\infty$. What exactly is it?

The (up to isomorphism) infinite cyclic group is just $\mathbb{Z}$ under addition.
You can visualize it as the group of integer shifts of the integers.
You can also visualize it as the rotations of the circle through integer numbers of radians, but that's not pretty geometrically since the orbit of any point on the circle is dense.
The group of all rotations of the circle is not cyclic.
