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## Group Multiplication Table

Let us consider the symmetry group of molecule. These are:
• a 3-fold axis, associated with two symmetry operations: (+120 rotation) and (-120 rotation).
• 3 vertical planes, , , and associated with tree mirror reflections.

Now we add to these five symmetry operations the identity operator and show that all six symmetry operations joint a group. Particularly, it is easy to see that , where the identity operation can be considered as a "product" of the two rotation operators operations and . It can also be seen that and . Following this procedure we can build the "multiplication table" presented below.

0.4cm

According to the Table 1, the "product" of each two symmetry transformations from six , , , , , and is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these six operators build a group. This group is known as group. The total number of operations in a group is called the group order. Therefore, the order of is 6.

Let us consider the symmetry group of molecule. The symmetry elements are:

• a 2-fold axis, associated with the symmetry operation: .
• two vertical planes, , and associated with two mirror reflections.

Now we add to these four symmetry operations the identity operator and show that all six symmetry operations joint a group. The "multiplication table" presented below.

0.4cm

According to the Table 2, the "product" of each two symmetry transformations from six , , , and is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these four operators build a group. This group is known as group. The group order of is 4.

Each point group is characterized by each own multiplication table.

Next: Some Consequences of Molecular Up: The Symmetry Classification of Previous: Point Groups   Contents
Markus Hiereth 2005-02-09