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Math Help - Cosets

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    Cosets

    What are the left cosets of



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  2. #2
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    Quote Originally Posted by Nightfly View Post
    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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