# Group of Rotations about Fixed Point is not Abelian

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## 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 $O$ 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:

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: $\text{(iii)}$