I have one question about group theory.
Our professor told us that there is a standard type of problems when it comes to proving that two groups are isomorphic. And he was always using the First theorem of homomorphism to prove them. But in our exam they gave us such problems, without teaching us the theorem and so I tried to prove them by constructing some images from one group to the other.
But my question is: Can we prove that, for example, $H/\mathbb{R}_+ \cong \mathbb{C}_9$ for $H=\{ z ∈ \mathbb{C}^* :|z|^9 = z^9\} $ without this theorem?
Thanks in advance!