the group of proper orthogonal transformations SO(3) acts transitively on the 2-sphere S(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}.

show that the isotropy group of any vector r is isomorphic to SO(2) and find a bijective correspondence between the factor space SO(3)/SO(2) and the 2-sphere such that SO(3) has identical left actions on these two spaces.

for the first part i have to show that the set of all 3x3 orthogonal matrices A such that Ar = r is isomorphic to the set of all 2x2 orthogonal matrices. so each matrix in SO(3) represents a rotation of some kind. but it seems to me that the only 3x3 orthogonal matrices that satisfy Ar = r are those that represent 360 degree rotations around some axis and it seems to me that set of all 2x2 orthogonal matrices contains more than just full rotations around some axis. so i am having trouble coming up with a bijection between the two sets.

for the second part i am confused about what they mean by identical left actions. i think they mean the same function on both which makes sense since we will be showing that SO(3)/SO(2) and the 2 sphere are in bijection. but doesn't it have to be an isomorphism as well to consider them to be the same group? i am also having trouble coming up with a bijection for the second part. could someone offer some hints to help me continue with this problem? thanks!

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# Left group actions involving SO(3) and the 2-sphere

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