D 2
E
C 2x
C 2y
C 2z`
E
E
C 2x
C 2y
C 2z
C 2x
C 2x
E
C 2z
C 2y
C 2y
C 2y
C 2z
E
C 2x
C 2z
C 2z
C 2y
C 2x
E
C 2v
E
C 2
r v1
r v2
E
E
C 2
r v1
r v2
C 2
C 2
E
r v2
r v1
r v1
r v1
r v2
E
C 2
r v2
r v2
r v1
C 2
E
By a close look at the group multiplication tables, a one-to-one correspondence
is observed clearly. Hence, they are isomorphic.
Finite Group
A group containing finite number of elements is called a finite group. For example,
crystallographic point groups and space groups are finite groups.
Generators of a Finite Group
It is possible to generate all elements of a group by starting from a certain set of
elements (at the most three) and taking their power and products. However, it is to
be noted that the definition of generator is not always unique. For example, in the
point group (D 2 ) 222, the possible generator are 2[100] and 2[010] or 2[100] and 2
[001] or 2[010] and 2[001].
Subgroups and Super Groups
A set of symmetry elements is said to be a subgroup of a bigger group (called super
group) if the set itself forms a group and satisfies all group conditions. In general,
every group has two trivial subgroups, the identity element and the group itself.
However, in the simplest term, it can be said that the addition of symmetry elements
to a point group produces super groups while the suppression of the symmetry
elements from point group produces subgroup. For example, we know that the point
group E(1) is the least symmetric and is the subgroup of all other 31 point groups.
On the other hand, the point groups D 6h (6/mmm) and O h (m3m) can have no super
group because no symmetry elements could be added to them to obtain any new
point group. Subgroups, super groups and order of the point groups are illustrated in
Fig. 6.24.
A group is called proper group if there are symmetry elements of the super group
not contained in the subgroup. For example, the set of point E (1), C 2 (2), r h
(m) and S 2 ( 1) is a proper subgroup of the point group C 2h (2/m).
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6 Unit Cell Symmeteries and Their Representations
E
C 2x
C 2y
C 2z`
E
E
C 2x
C 2y
C 2z
C 2x
C 2x
E
C 2z
C 2y
C 2y
C 2y
C 2z
E
C 2x
C 2z
C 2z
C 2y
C 2x
E
C 2v
E
C 2
r v1
r v2
E
E
C 2
r v1
r v2
C 2
C 2
E
r v2
r v1
r v1
r v1
r v2
E
C 2
r v2
r v2
r v1
C 2
E
By a close look at the group multiplication tables, a one-to-one correspondence
is observed clearly. Hence, they are isomorphic.
Finite Group
A group containing finite number of elements is called a finite group. For example,
crystallographic point groups and space groups are finite groups.
Generators of a Finite Group
It is possible to generate all elements of a group by starting from a certain set of
elements (at the most three) and taking their power and products. However, it is to
be noted that the definition of generator is not always unique. For example, in the
point group (D 2 ) 222, the possible generator are 2[100] and 2[010] or 2[100] and 2
[001] or 2[010] and 2[001].
Subgroups and Super Groups
A set of symmetry elements is said to be a subgroup of a bigger group (called super
group) if the set itself forms a group and satisfies all group conditions. In general,
every group has two trivial subgroups, the identity element and the group itself.
However, in the simplest term, it can be said that the addition of symmetry elements
to a point group produces super groups while the suppression of the symmetry
elements from point group produces subgroup. For example, we know that the point
group E(1) is the least symmetric and is the subgroup of all other 31 point groups.
On the other hand, the point groups D 6h (6/mmm) and O h (m3m) can have no super
group because no symmetry elements could be added to them to obtain any new
point group. Subgroups, super groups and order of the point groups are illustrated in
Fig. 6.24.
A group is called proper group if there are symmetry elements of the super group
not contained in the subgroup. For example, the set of point E (1), C 2 (2), r h
(m) and S 2 ( 1) is a proper subgroup of the point group C 2h (2/m).
246
6 Unit Cell Symmeteries and Their Representations