In group theory, geometry, representation theory and molecular geometry, 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, as transformations of an object in space, rotations, reflections and inversions are all symmetry operations. Such symmetry operations are performed with respect to symmetry elements (for example, a point, line or plane). 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.