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# make this into a group. watch

1. take the unit circle.

let's have a set X of elements (a,b) s.t. a and b are on the boundary of the circle, and are distinct.

I want to come up with some operation that makes X into a group. But I cannot preserve the distinctness! any ideas?
2. I think the following is right, though I haven't checked the details.

Essentially, X is the torus [0,1]^2 with the diagonal {(x,x)} removed. If you draw this you'll notice it splits into two triangles, but opposite edges of the square are identified. So you might as well move the two triangles so that two identified edges are together. Now you have a parallelogram with the short edges identified and the long edges 'open'. This is just a copy of (0,1) * circle, both of which are groups. ((0,1) is easily seen to be a group by homomorphically mapping it to R - use tan.)

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Updated: February 13, 2010
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