Cosets

**Nightfly** · December 9th 2008, 02:39 PM

What are the left cosets of

Thanks.

**ThePerfectHacker** · December 9th 2008, 03:22 PM

Originally Posted by Nightfly

What are the left cosets of

Let $\alpha,\beta \in \mathbb{C}^{\times}$ . We have that $\alpha H = \beta H \implies \alpha \beta^{-1} \in H \implies |\alpha \beta^{-1}| = 1 \implies |\alpha| = |\beta|$ .
Geometrically two points in $\mathbb{C}^{\times}$ are in the same coset if and only if they lie on the same circle.
Using our intuition it seems that $\mathbb{C}^{\times}/H \simeq \mathbb{R}^+$

We can prove this by defining $\phi : \mathbb{C}^{\times} \to \mathbb{R}^+$ by $\phi (z) = |z|$ .
It follows that $\phi [\mathbb{C}^{\times}] = \mathbb{R}^+$ and $\ker (\phi) = \{ z\in \mathbb{C}^{\times} : |z| = 1 \} = H$ .
Therefore, by fundamental homomorphism theorem we have $\mathbb{C}^{\times}/H \simeq \mathbb{R}^+$ .

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