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I want to prove that the setSO(2)of orthogonal 2x2 matrices with det=1 is a Lie group.

The group operation is of course assumed to be the ordinary matrix multiplication [itex]\times[/itex]:SO(2)→SO(2).

I made the following attempt but then got stuck at one point.

We basically have to prove that SO(2) is adifferentiable manifold.

In order to do so I consider the group isomorphism [tex]\left( SO(2), \times \right) \cong \left( [0,2\pi), +_{\mathrm{mod\, 2\pi}} \right)[/tex] where +_{mod 2∏}means addition modulo 2∏.

At this point I should prove that [itex]f(x,y) = (x+y)\, \mathrm{mod}\, 2\pi[/itex] is smooth, and that [itex]f^{-1}(x) = (2\pi -x)\, \mathrm{mod}\, 2\pi[/itex] is also smooth.

This is where I get stuck.

For instance, [itex]f^{-1}[/itex] is certainly smooth in the open interval (0,2∏), but it is not differentiable at x=0, because at x=0 we don't have an open neighborhood where the [itex]f^{-1}[/itex] would be defined.

How do we address this issue?

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# Help needed to prove that SO(2) is a Lie group

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