A class consists of elements of a group that are conjugate to one another. Two elements are conjugate to each other if , where . If is conjugate to , then is conjugate to , as

where (note that according to the inverse property of a group). The identity element is a class by itself since .

If the elements of are represented by matrices, is called a similarity transformation. Furthermore, if and are conjugate to each other, and and are conjugate to each other, then and are conjugate to each other. This is because and and therefore, , where .

###### Question

Show that the symmetry operators , and of the point group belong to the same class.

###### Answer

With reference to the multiplication table,

we have

Similarly, . Since and are conjugate to each other, and and are conjugate to each other, then and are conjugate to each other. Therefore, form a class. Using the same logic, we find that and form another class.

All elements of the same class in a group have the same order, which is defined as the smallest value of such that , where . This is because if is conjugate to , we have

The above equation of is valid if and only if . This means that the smallest value of in and in must be the same. Therefore, elements and of the same class in a group have the same order and is denoted by .

###### Question

Verify that the symmetry operators , and of the point group have the same order of 2.

###### Answer

It is clear that when the reflection operator acts on a shape twice, it sends the shape into itself. The same goes for and . Hence, .

As mentioned in an earlier article, the similarity transformation of a matrix to a matrix leaves the trace of , which is defined as , invariant. This implies that elements of the same class in a group have the same trace.