Trying to prove that the proper orthogonal group SO(3) acts transitively on the set of points on the surface of a sphere. Can show that if you assume 2 points on the sphere can be related by x=Ay where A is a 3x3 matrix then A multiplied by its transpose must be I but this only shows A would be in O(3) not SO(3) - i.e. how can you show any 2 points can be linked by A where det(A) = 1?

 

Any help would be much appreciated!

You could appeal to the following theorem.

 

Theorem

A Lie group [math]G[/math] acts globally and transitively on an manifold [math]M[/math] if and only if [math]M\simeq G/H[/math] is isomorphic to the homogeneous space obtained by quotienting [math]G[/math] by the isotropy subgroup [math]H= G_{x} [/math] of any designated point [math]x\in M[/math].

 

So let [math]G = SO(3)[/math] be a group acting on [math]R^{3}[/math]. Let [math]H = SO(2)[/math] the isotropy subgroup of a point [math]x \in R^{3}[/math].

 

The group [math]SO(3)[/math] acts on [math]S^{2}[/math] transitively as [math]S^{2} = SO(3)/SO(2)[/math].

 

 

 

Note that [math]SO(2)[/math] is not a normal subgroup of [math]SO(3)[/math] and so [math]S^{2}[/math] does not have a group structure.