Group Theory is a branch of the mathematical field of algebra. One important application, the theory of symmetry groups, is a powerful tool for the prediction of physical properties of molecules and crystals. It is for example possible to determine whether a molecule can have a dipole moment. Many important predictions of spectroscopic experiments (optical, IR or Raman) can be made purely by group theoretical considerations. The qualitative properties of molecular orbitals can be obtained from group theory (whereas their precise energetics and ordering have to be determined by a quantum chemical method). In quantum chemistry, group theory can applied to ab initio or semi-empirical calculations to significantly reduce the computational cost. 7.1: Introduction 7.2: Symmetry Operations and Elements 7.3: Symmetry Elements A symmetry operation is an action that leaves an object looking the same after it has been carried out. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out. The symmetry element consists of all the points that stay in the same place when the symmetry operation is performed. 7.4: Successive Operations 7.5: Point Groups 7.6: Character Tables - An Introduction Although the method of assigning a point group to a molecule depends on some knowledge of the symmetry elements the molecule has, it does not require the consideration of all elements. This is where the character table of the point group comes into play.