Symmetry operation

In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1⁄3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element (a point, line or plane).^[1]

In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness.

1. Physical properties must be invariant with respect to symmetry operations.
2. Symmetry operations can be collected together in groups which are isomorphic to permutation groups.

In the context of molecular symmetry, quantum wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.