A group is a complete set of symmetry operations that describes the symmetry of a particular object. In chemistry we use groups to classify molecules according to their symmetry. When we say that the set of operations is complete we mean that any two operations carried out successively will generate another operation of the group. We can collect this set of operations and use it as a basis for a mathematical space that defines the properties of the molecule. A group has a dimension that is equal to the number of symmetry operations. A groups also has a set of basis vectors that define the space. These are called irreducible representations.
Once we have identified sets of symmetry operations that comprise a group we need to establish a procedure for assignmeent of molecules to the various point groups. Of course, not all molecules have symmetry. Molecules that lack and symmetry belong to a group known as C1. If the molecule only has mirror symmetry it belongs to Cs. It is rare, but some molecules only have an inversion center and therefore belong to Ci. The most common and useful groups are those with a rotaional axis. The groups that have only one rotational axis are the C groups. Ckv have an n-fold rotational axis and n verical mirror planes. Cnh have an n-fold axis and one horizontal mirrir plane. THe D groups have an n-fold axis and n perpendicular 2-fold axes. Each os this groups is discussed in the section below.
