Unformatted text preview: 59 2.77 (i) Prove that the special orthogonal group SO ( 2 , R ) is isomorphic to the circle group S 1 . Solution. De fi ne Ï• : A = Â· cos Î± âˆ’ sin Î± sin Î± cos Î± Â¸ 7â†’ ( cos Î±, sin Î±). In Example 2.48(iii), the product of two special orthogonal matri ces is computed, and this shows that the function Ï• is a homomor phism. The inverse function sends ( cos Î±, sin Î±) to the matrix in the de fi nition of Ï• . (ii) Prove that all the rotations of the plane about the origin form a group under composition which is isomorphic to SO ( 2 , R ) . Solution. By Proposition 2.59, every isometry fi xing the origin is a linear transformation. The usual isomorphism between linear transformations and matrices (done in general in Chapter 4) exists here. If Â² 1 = ( 1 , ), Â² 2 = ( , 1 ) is the standard basis of R 2 , then Ï• corresponds to the 2 Ã— 2 matrix whose fi rst column are the coor dinates of Ï•(Â² 1 ) and whose second column are the coordinates of Ï•(Â² 2 ) , as in Example 2.48(iii)., as in Example 2....
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 Fall '11
 KeithCornell
 Group Theory, Normal subgroup, Abelian group

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