Let $\mathbb{C}^{*}$ denote the group of nonzero complex numbers under multiplication, and $S^{1} \subset \mathbb{C}^{*}$ the subgroup of complex numbers of length one. Torsion elements of $\mathbb{C}^{*}$ are called roots of unity.
Show that $\text{Tor}(\mathbb{C}^{*}) \subset S^1.$ Now give a simple reason that $\text{Tor}(\mathbb{C}^{*}) \neq S^1.$
My question is:
1- I know from here Torsion subgroup of $\mathbb{C}^\times$ that the torsion elements are the roots of unity, but I do not know how to prove that $\text{Tor}(\mathbb{C}^{*}) \subset S^1.$ could anyone help me in writing a rigorous proof for that, please?
2- What is a simple reason that $\text{Tor}(\mathbb{C}^{*}) \neq S^1$?
EDIT:
My definition of $S^{1}$ is $\{ z \in \mathbb{C^{*}\ :\ |z|=1 }\}$
Best Answer
Here is an algebraically concise way to answer your questions: