### Identify the quotient group (NBHM-2014)

Let $\mathbb{C}^{*}$ denote the multiplicative group of non-zero complex numbers and let $\mathbb{R}^*$ denote the subgroup of positive real numbers. Identify the quotient group.

Solution :

Consider the function $f:\mathbb{C}^* \to \mathbb{C}^*$ defined by $f(r.e^{i\theta})= e^{i\theta}$  then $f$ is a homomorphism with kernel $\mathbb{R}^*$.

Therefor by the fundamental theorem of homomorphism, we can say that $\mathbb{C}^*/\mathbb{R}^*$ is isomorphic to the image $f(\mathbb{C}^*)=S^1$.

where $S^1=\{z\in\mathbb{C} | |z|=1\}$. Note that $f(z) = \frac{z}{|z|}$ is the above given map.

The fundamental theorem of homomorphism: Check my post regarding this for a wonderful article.

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