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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?

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    $\begingroup$ 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. $\endgroup$ – Nate Eldredge Jul 21 '19 at 14:47
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    $\begingroup$ 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. $\endgroup$ – Fakemistake Jul 21 '19 at 14:59
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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.

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