Group of Rotations about Fixed Point is not Abelian

Theorem
Let $\mathcal S$ be a rigid body in space.

Let $O$ be a fixed point in space.

Let $\mathcal G$ be the group of all rotations of $\mathcal S$ around $O$.

Then $\mathcal G$ is not an abelian group.

Proof
Let $\mathcal S$ be a square lamina.

Let $O$ be the center of $\mathcal S$.

Recall the definition of the symmetry group of the square $D_4$:

We have that:


 * Reflection $t_x$ can be achieved by a rotation of $\mathcal S$ of $\pi$ radians about $x$.


 * Reflection $t_y$ can be achieved by a rotation of $\mathcal S$ of $\pi$ radians about $y$.

Thus $D_4$ forms a subgroup of $\mathcal G$.

From Symmetry Group of Square is Group we have that $D_4$ is not abelian.

From Subgroup of Abelian Group is Abelian it follows that $\mathcal G$ is also not abelian.