SO(3)

Overview

$SO(3)$ is the rotation group of ${\mathbb{R}}^3$

Subgroups

Other than the obvious subgroup $SO(2)$, there are also three other subgroups called the polyhedral groups (induced by the rotations that map regular polyhedron into itself)

They are:

• Tetrahedral group ($T$) - order 12
• Octahedral group ($O$) - order 24
• Icosahedral group ($I$) - order 60

The order of these groups can be calculated in the following way: a position of a such polyhedron is uniquely determined by:

• Which of the faces is in the position of the front face
• Which of the edges of that face is at the bottom of the front face

Therefore, we get for example $|T|=$ 4 faces $\times$ 3 $=12$ edges each face

Simplicity

SO(3) is a simple group

SO(3) is a simple group The only nontrivial subgroup of SO(3) closed under conjugation is SO(3) itself

Proof: suppose $h$ is a nontrivial rotation of a nontrivial normal subgroup $H$, then $H$ includes $g^{-1}hg\forall g\in SO(3)$. Let’s say $h$ rotate about $l$ through angle $\alpha$ Consider the geometric meaning of this conjugate, $g$ can be seen as a transformation of the rotation axis, then $H$ must contain all possible rotations through angle $\alpha$

Now consider the composition of two rotations with angle $\alpha$ (see Reflection representation of isometries). This composition can be seen as a rotation through angle $\theta$ We see that by moving $P$ we can obtain rotations with any angle ($\theta \in [0,2\pi )$)