32
3 Groups
elements, etc. Each time a coset is formed, a block of size |H | is occupied till the
full territory of the group is occupied by subsets of the same size. Their order must
thus be a divisor of the group order. In a sense, one could describe this collection
of cosets as the quotient, resulting from the “division” of the group by a subgroup.
We shall make use of this concept in the induction of representations in Sect. 4.6.In
the present example, the point group of ammonia is of order 6, with divisors 1, 2, 3,
and 6, and for each of these, there are indeed subgroups. This is rather exceptional,
though. It is not the case that for every divisor there should be a subgroup. As a
further corollary, groups with an order which is a prime number have no nontrivial
subgroups.
3.6 Classes
Probably, the most natural way to partition a group is by putting all elements “of the
same kind” into separate classes. Hence, in the group C 3v we could put the three
planes in one class; the unit element is of course a separate class, but what about
the ˆ
C 3 and ˆ
C 2
3 axes? Are they of the same kind or not? Clearly, we need a rigorous
definition of what it means for two symmetry elements to be equivalent. We can use
as a criterion the symmetry transformations of an operator, as explained in Sect. 1.3.
Hence, two elements ˆ
A and ˆ
B will belong to the same class, ˆ
A ∼ ˆ
B, if there exists
(denoted as ∃) a symmetry operation ˆ
U that belongs to the group and transforms ˆ
B
into ˆ
A:
ˆ
A ∼ ˆ
B :∃ ˆ
U ∈ G → ˆ
A = ˆ
U ˆ
B ˆ
U
−1
(3.24)
In C 3v , the symmetry planes will map ˆ
C 3 onto ˆ
C 2
3 . Hence, we can safely say that the
two threefold elements belong to the same class, because of the existence of symmetry planes, which can reverse the direction of rotation. Synonyms for “to belong
to the same class” are: “to be (class-)conjugate” or “to be similarity transforms.” If
the element ˆ
U transforms ˆ
B to ˆ
A, then the inverse element, ˆ
U −1 , which because
of the group properties also belongs to G, will do the reverse and will transform ˆ
A
into ˆ
B. Hence, conjugation is reflexive. It is, furthermore, transitive:
( ˆ
A ∼ ˆ
B)&( ˆ
B ∼ ˆ
C) → ( ˆ
A ∼ ˆ
C)
(3.25)
The set of all elements that are conjugate with a given element is a conjugacy class
or simply a class. A class is fully denoted by specifying any one of its elements in
the same way as a coset is defined by any one of its representatives. The total group
is of course the sum of all its classes. In abelian groups, the similarity transformation
will always return the element on which it was acting; hence, in this case, all classes
will be singletons (sets of order 1). The unit element is unique, so it is always in its
own class.
Theorem 3 The number of elements in a class is a divisor of the group order.
3 Groups
elements, etc. Each time a coset is formed, a block of size |H | is occupied till the
full territory of the group is occupied by subsets of the same size. Their order must
thus be a divisor of the group order. In a sense, one could describe this collection
of cosets as the quotient, resulting from the “division” of the group by a subgroup.
We shall make use of this concept in the induction of representations in Sect. 4.6.In
the present example, the point group of ammonia is of order 6, with divisors 1, 2, 3,
and 6, and for each of these, there are indeed subgroups. This is rather exceptional,
though. It is not the case that for every divisor there should be a subgroup. As a
further corollary, groups with an order which is a prime number have no nontrivial
subgroups.
3.6 Classes
Probably, the most natural way to partition a group is by putting all elements “of the
same kind” into separate classes. Hence, in the group C 3v we could put the three
planes in one class; the unit element is of course a separate class, but what about
the ˆ
C 3 and ˆ
C 2
3 axes? Are they of the same kind or not? Clearly, we need a rigorous
definition of what it means for two symmetry elements to be equivalent. We can use
as a criterion the symmetry transformations of an operator, as explained in Sect. 1.3.
Hence, two elements ˆ
A and ˆ
B will belong to the same class, ˆ
A ∼ ˆ
B, if there exists
(denoted as ∃) a symmetry operation ˆ
U that belongs to the group and transforms ˆ
B
into ˆ
A:
ˆ
A ∼ ˆ
B :∃ ˆ
U ∈ G → ˆ
A = ˆ
U ˆ
B ˆ
U
−1
(3.24)
In C 3v , the symmetry planes will map ˆ
C 3 onto ˆ
C 2
3 . Hence, we can safely say that the
two threefold elements belong to the same class, because of the existence of symmetry planes, which can reverse the direction of rotation. Synonyms for “to belong
to the same class” are: “to be (class-)conjugate” or “to be similarity transforms.” If
the element ˆ
U transforms ˆ
B to ˆ
A, then the inverse element, ˆ
U −1 , which because
of the group properties also belongs to G, will do the reverse and will transform ˆ
A
into ˆ
B. Hence, conjugation is reflexive. It is, furthermore, transitive:
( ˆ
A ∼ ˆ
B)&( ˆ
B ∼ ˆ
C) → ( ˆ
A ∼ ˆ
C)
(3.25)
The set of all elements that are conjugate with a given element is a conjugacy class
or simply a class. A class is fully denoted by specifying any one of its elements in
the same way as a coset is defined by any one of its representatives. The total group
is of course the sum of all its classes. In abelian groups, the similarity transformation
will always return the element on which it was acting; hence, in this case, all classes
will be singletons (sets of order 1). The unit element is unique, so it is always in its
own class.
Theorem 3 The number of elements in a class is a divisor of the group order.