I am currently dealing with some group theory problems. My algebra textbook denotes the unit circle on the complex plane by $S^1$.
I am pretty confused because when I searched the term circle group on Wikipedia, a notation $\mathbb{T}$ was used for a unit circle. It is said that $\mathbb{T}$ and $SO(2)$ are isomorphic, which makes sense for me. But I know that a torus is $S^1\times S^1$ and the group here is a symmetric group. So here is my question:
Does $S^1$ only mean a unit circle? Or does it have something to do with symmetric groups?