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 $$(\operatorname{mod} n)$$, 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 $$(\operatorname{mod} 2\pi)$$. 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 rotation group of a circle, which is what I interpreted $C_\infty$ to be". Well, there we are. – Lord Shark the Unknown May 25 at 15:36
• Is there a good geometric interpretation then? The best one I can think of is rotating a circle by an integer number of radians. – Ovinus Real May 25 at 15:39
• The order of the rotation group of a circle is $2^{\aleph_0}$. Whether or not this is $\aleph_1$ is not decidable from the axioms of set theory. – DanielWainfleet May 25 at 15:56
• Should have just said uncountable, sorry – Ovinus Real May 25 at 16:05

The (up to isomorphism) infinite cyclic group is just $$\mathbb{Z}$$ under addition.