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.
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:
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.
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.
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