Group of Rotations about Fixed Point is not Abelian

Theorem

Let $\SS$ be a rigid body in space.

Let $O$ be a fixed point in space.

Let $\GG$ be the group of all rotations of $\SS$ around $O$.

Then $\GG$ is not an abelian group.

Proof

Let $\SS$ be a square lamina.

Let $O$ be the center of $\SS$.

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

Let $\SS = ABCD$ be a square.

The various symmetry mappings of $\SS$ are:

the identity mapping $e$

the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively

the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively

the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$

the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square, and can be denoted $D_4$.

We have that:

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

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

Thus $D_4$ forms a subgroup of $\GG$.

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 $\GG$ is also not abelian.

$\blacksquare$

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(iii)}$