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How to show the circle group $$\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$$ is isomorphic to $\mathbb{C}^*/\mathbb{R}^+$?

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What is $C^*/R^+$ ? –  Amr Jan 7 '13 at 2:10
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Have you learned any theorems with quotients and isomorphisms in them? –  Jonas Meyer Jan 7 '13 at 2:11
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From context it appears that $\mathbb C^*=\mathbb C\setminus \{0\}$ with multiplication, and $\mathbb R^+$ is the subgroup of positive real numbers with multiplication, but I agree with @Amr that it wouldn't hurt to make this explicit. –  Jonas Meyer Jan 7 '13 at 2:13
    
@JonasMeyer I see. Just to make sure what is the circle group ? –  Amr Jan 7 '13 at 2:15
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@Amr: Many of these things can be looked up in Wikipedia: circle group. –  MJD Jan 7 '13 at 2:39
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up vote 6 down vote accepted

What about

$$f:C^*\to \Bbb T\,\,\,,\,\,f(z):=\frac{z}{||z||}\;\;?....$$

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..you are right...thank you! –  hxhxhx88 Jan 7 '13 at 3:46
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@hxhxhx88: You can also show that $\Bbb T\cong\frac{\Bbb R}{\Bbb Z}$ via $\phi:\Bbb R\to \Bbb T, \phi(x)=\text{e}^{2\pi x i}$. –  B. S. Jan 7 '13 at 6:56
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