- #1

quantum_smile

- 21

- 1

## Homework Statement

Take the subgroup isomorphic to SO(2) in the group SO(3) to be the group of matrices of the form

[tex]

\begin{pmatrix} g & & 0 \\ & & 0 \\ 0 & 0 & 1 \end{pmatrix}, g\in{}SO(2).

[/tex]

Show that there is a one-to-one correspondence between the coset space of SO(3) by this subgroup and the two-dimensional sphere

[tex]

SO(3)/SO(2)=S^2.

[/tex]

## Homework Equations

[tex]

SO(3)/SO(2)=\{[A] | A\in{}SO(3)\}\\

[A]:=\{B\in{}SO(3) | B = AH, H\in{}SO(2)\}

[/tex]

## The Attempt at a Solution

As a first step, I tried to get a more intuitive grasp of what SO(3)/SO(2) "is." However, all I've been able to get is a series of cumbersome matrix component equations, and I have yet to figure out how to really utilize the fact that all the matrices are orthogonal and that their determinants are 1.

Also, I've only learned very basic topology, so I don't know how to set up a one-to-one correspondence between SO(3)/SO(2) and S^2.

Any help/hints are very appreciated!