Group of Rotations about Fixed Point is not Abelian

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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 $D_4$:


Let $\mathcal S = ABCD$ be a square.

SymmetryGroupSquare.png

The various symmetry mappings of $\mathcal S$ are:

The identity mapping $e$
The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square.


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.

$\blacksquare$


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