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Math Help - complex number and images and kernel

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    complex number and images and kernel

    I have attached a problem here and my attempt-
    peobelm asks me to then identify quotentient group G/Ker (phi) up to isomorphism, I have found the kernel but not sure about Image.
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    Re: complex number and images and kernel

    You have correctly found the kernel i.e. \ker \phi=(\mathbb{R}-\{0\})i. Now, if w\in\textrm{Im}\;\phi then, there exists z\in\mathbb{C}-\{0\} such that w=z/\bar{z}. This implies |w|=|z/\bar{z}|=|z|/|\bar{z}|=1. Could you continue?
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    Re: complex number and images and kernel

    Not really as I have tried looking on net and my struggles are clear. If you can show me this time around I'll be comfortable with similar problems if you don't mind☺thanks
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    Re: complex number and images and kernel

    I think it's R+ but don't know how to mathematically get there
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    MHF Contributor FernandoRevilla's Avatar
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    Re: complex number and images and kernel

    Quote Originally Posted by mathslearner View Post
    I think it's R+ but don't know how to mathematically get there
    It is the unit circle S_1. If w\in S_1 then, w=\cos \theta +i\sin \theta for some \theta \in [0,2\pi]. i.e. we can express

    w=\dfrac{\cos (\theta/2)+i\sin (\theta/2)}{\cos (-\theta/2)+i\sin (-\theta/2)}=\dfrac{z}{\bar{z}}\in \textrm{Im}\;\phi.
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