# 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.

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:

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.

$\blacksquare$

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 3$: Examples of Infinite Groups: $\text{(iii)}$