Two-dimensional inversion symmetry as the fundamental symmetry of incompressible quantum Hall fluids

Two dimensional inversion symmetry (${180}^{\circ }$ rotations in the ``Hall plane’’ that hosts the incompressible electron fluid that exhibits the quantized Hall effect) is identified as its fundamental unbroken symmetry. A consequence is that the integers $p$ and $q$ which define both the Landau level filling factor $\nu$ = $p\mathrm{/}q$ and the elementary fractional charge $\pm e\mathrm{/}q$ of topological excitations, cannot have a common divisor greater than 2.