# SO(3)

The lie group of all 3D rotations of Euclidean space \(\mathbb{R}^3\) where the group (algebra) operation is composition.

## 1. relation with SU(2)

- There is a surjective 2-to-1 homomorphism from SU(2) and SO(3) where SU(2) is the group of all unitary matrix with determinant 1
- There is an isomorphism from \(PU(2)\) to \(SO(3)\) see excellent stack overflow where \(PU(2)\) is the set of all unitary matrices that are identified up to a global phase scaling factor
- \(PU(2)\) acts on \(\mathbb{R}^3\) via \(SO(3)\)
- see here for difference between \(SU(2)\) and \(PU(2)\)
- just to be clear, \(SU(2)\) is not the space of all matrices that can be quantum gates. That is \(U(2)\) and the differences up to scalar multiples are not physically measurable