so the question is: Define multiplication on $\mathbb{R}^2\setminus(0,0)$ s.t. $\mathbb{R}^2\setminus(0,0)/\text{SO}_2(\mathbb{R}) \approx \mathbb{R}^2_{\gt0}$.

I know that $\mathbb{R}^2\setminus(0,0)$ isomorphic to $\mathbb{C}^*$, and the multiplication is the complex multiplication, and the $SO_2$ group is isomorphic to $S^1$, and $S^1$ is subgroup of $C^*$, but now I'm stuck. I don't understand what are the elements in the quotient group, and how to make isomorphism to $\mathbb{R}_{\gt0}$

Thank you very much!


Hint: if you take any number $(r,\theta)$ in polar coordinates in the punctured plane and quotient off by the rotations, all that is left is a coset $[(r,\theta)]$ which does not depend on the angle $\theta$ and might as well be represented by the modulus $r$ alone. Therefore the obvious isomorphism would map such a coset to $r \in \mathbb{R}_{>0}$, hinting on the fact that the group operation you should use in the punctured plane is complex multiplication $(r,\theta)(r',\theta') = (rr',\theta + \theta')$.

  • $\begingroup$ if i understand you correctly, you say i can think of the quotient group as all the circles and i just take their radius? $\endgroup$ – נוי רחמני Nov 29 '17 at 13:27
  • $\begingroup$ @נוירחמני Yes, geometrically each coset $[(r,\theta)]$ is just the circle centered at the origin with radius $r$. $\endgroup$ – Alex Provost Nov 29 '17 at 13:29

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