I'm trying to use the First Isomorphism Theorem to show that $\mathbb{T}^1 \cong \mathbb{C}^*/\mathbb{R}_{&gt;0}$ by constructing a surjective group homomorphism from the nonzero complex numbers to the circle group whose kernel is the set of positive reals. I haven't yet taken a course in complex variables, so I did some digging around and could only find a homomorphism from $\mathbb{R}^*$ to $\mathbb{T}^1$ defined by $f(\theta) = e^{i\theta}$. I don't really know what that means, so I'm struggling to construct any homomorphism $\phi : \mathbb{C}^* \to \mathbb{T}^1$, let alone one whose kernel is $\mathbb{R}_{&gt;0}$.  EDIT: Does $f(z) = \frac{z}{|z|}$ work? EDIT2: Probably not. I think the image of this is actually just $\{\pm 1\}$. I really have no idea how to work with complex numbers at all.
Let $G = (\mathbb{C} - \{0\}, \cdot)$, a subgroup $U = \{x+yi \mid x^2 + y^2 = 1\}$. Use the Fundamental Theorem to show that $G/(\mathbb{R}_{&gt; 0},\cdot)$ is isomorphic to $U$ where $\mathbb{R}_{&gt; 0}$ denotes the positive reals.  I am struggling to find the correct mapping from $G$ to $U$ that results in the the kernel being the nonnegative reals under multiplication. I'm hoping to get help with this mapping then I can show it is a homomorphism and onto and can use the Fundamental Theorem.
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