There are some conclusions that may be drawn about a molecule as soon as its point group has been identified:

A polar molecule is one which possesses a permanent electric dipole moment.

A molecule belonging to a group Cn can only have a dipole moment parallel to the rotational axis.

This is because the symmetry of the molecule implies that any component of a dipole moment perpendicular to the axis must be exactly cancelled out by opposing components. (There must be opposing components around the axis to maintain the rotational symmetry of the molecule.) However, since the group contains no operations that interchange the two “ends” (as defined by the rotational axis) of the molecule, it is possible for a dipole moment to exist along this axis. The same is true of the groups Cnv. For the majority of the other groups, there are operations which do exchange the two ends of the molecule (horizontal reflections, improper rotations, perpendicular C2 axes or inversions), and these operations imply that any dipole moment along the axis is cancelled out by an exactly equal dipole moment pointing in the other direction along the axis.

The upshot of such considerations is that we may conclude that molecules may only have a dipole moment if they belong to the groups Cn , Cnv , Cs or C1 . In the first two groups, the dipole must lie along the rotational axis. In a molecule belonging to Cs, the dipole must lie in the in the mirror plane.

Note being a member of one of these groups does not require a molecule to be polar, it merely permits it.

Symmetry also gives information about the chirality of molecules. A chiral molecule is one which cannot be superimposed upon its own mirror image, and they have the property of being optically active. That is, they rotate the plane of plane-polarised light.

From the theory of optical activity, it may be shown that a molecule can only be chiral if it does not possess an axis of improper rotation, Sn. It is important to recognise that axes of improper rotation may be present under a different name. An inversion centre, for example, is exactly the same as an S2 axis. Similarly a mirror plane is the same as an S1 axis. Note that the groups Cnh are also excluded, as they possess both Cn and σh, which are the two components of an Sn axis.

In summary, the only groups that may be chiral are as follows: C1, Cn, and Dn. Again, not all molecules in these groups will necessarily be chiral, they are merely permitted to be. For example, hydrogen peroxide belongs to the group C2, but it is not chiral, as free rotation about the O-O bond is possible.