Classes
Two elements A and B in a group belong to the same class if there is an element X
within group such that
X
À1 AX ¼ B
ð6:14Þ
where X
−1 is the inverse of X. From Eq. 6.14, we can say that B is a similarity
transform of A by X, or that A and B are conjugate to one another. Making use of
the similarity transform of one element by other elements one can determine
whether a set of elements from classes or not.
The order of a class (c) of the group must be an integral factor of the order of the
group (g), that is,
g ¼ mc
ð6:15Þ
The similarity transformation method is sometimes too elaborate to find the
classes, particularly in high symmetry systems. Therefore, an alternative method
Fig. 6.24 Illustration of subgroups, super groups and order of the point groups
6.3 Group (Point) Representation of Symmetry Operations
247
Two elements A and B in a group belong to the same class if there is an element X
within group such that
X
À1 AX ¼ B
ð6:14Þ
where X
−1 is the inverse of X. From Eq. 6.14, we can say that B is a similarity
transform of A by X, or that A and B are conjugate to one another. Making use of
the similarity transform of one element by other elements one can determine
whether a set of elements from classes or not.
The order of a class (c) of the group must be an integral factor of the order of the
group (g), that is,
g ¼ mc
ð6:15Þ
The similarity transformation method is sometimes too elaborate to find the
classes, particularly in high symmetry systems. Therefore, an alternative method
Fig. 6.24 Illustration of subgroups, super groups and order of the point groups
6.3 Group (Point) Representation of Symmetry Operations
247