# Cosets

• Dec 9th 2008, 02:39 PM
Nightfly
Cosets
What are the left cosets of
http://img520.imageshack.us/img520/2347/mathsnk4.png

Thanks.
• Dec 9th 2008, 03:22 PM
ThePerfectHacker
Quote:

Originally Posted by Nightfly

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

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