# Notation of a Unit Circle: Does $S^1$ only mean a unit circle? Or does it have something to do with symmetric groups?

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?

• Usually $S^n$ denotes an $n$-dimensional sphere, and $\mathbb{T}^n$ an $n$-dimensional torus. It so happens that the 1-dimensional sphere and the 1-dimensional torus are both the same object, namely a circle, and that the group of rotations of $\mathbb{R}^2$ can also be identified with a circle. – Nate Eldredge Jul 21 '19 at 14:47
• Be aware of superscript and subscript. $S^n$ is a sphere, while $S_n$ is for many authors the symmetric group of a set with $n$ elements. – Fakemistake Jul 21 '19 at 14:59

The notion $$S^1$$ refers to the unit circle and is mostly used in topology. That is because the unit circle is the $$1$$-dimensional sphere. More generally, $$S^k$$ is the $$k$$-dimensional sphere and $$T^2 = S^1 \times S^1$$ is the torus. If you are studying group theory or other topics you might find other notions like $$\mathbb{T}$$. Another common one would be $$U(1)$$. As a group, I would probably denote the torus by $$(\mathbb{R}/\mathbb{Z})^2 = \mathbb{T}^2$$ rather than taking products of $$S^1$$ since that reflects more the group theoretical aspect than the topological aspect.