26
3 Groups
Fig. 3.3 Cayley graph of the
C 3v point group. The
generators are c = ˆ
C 3 and
s =ˆ σ 1
A presentation of a group is a set of generators, together with a minimal set of relations that are sufficient to work out any product of two elements. As an example, let
us denote the ˆ
C 3 , ˆ
σ 1 generators as c, s. Just three relations among these generators
are sufficient to derive the whole multiplication table: c 3 = s 2 = e, sc = c 2 s.T h e
generation of the six elements of the group follows from Eq. (3.11):
ˆ
E = e
ˆ
C 3 = c
ˆ
C
2
3 = c
2
ˆ
σ 1 = s
ˆ
σ 2 = sc
ˆ
σ 3 = cs
(3.13)
Any product of these elements can be shown to be closed using only the presentation. As an example, the rule ˆ
σ 2 ˆ
σ 3 = ˆ
C 3 , is expressed in the presentation as follows:
sccs = s
c
2 s
= s(sc) = s
2 c = c
(3.14)
In this way the whole multiplication table can be derived.
The structure of the group can also be encoded in a graph known as the Cayley
graph. A graph is an abstract mathematical object consisting of a set of points, or
nodes, and a set of lines connecting pairs of these points. In a directed graph these
pairs are ordered, which means that directional arrows are added to the connecting
lines. In the Cayley graph every element of the group corresponds to a node. The
lines correspond to the action of the group generators. The generator ˆ
g connects
a given node ˆ
R i by a directed line to the resulting node ˆ g ˆ
R i . The action of
the group on its own Cayley graph will not only map nodes onto nodes, but will
also preserve the directed connections. As a result, the symmetry group will map
the graph onto itself. Such a mapping is called an automorphism. The group G is
3 Groups
Fig. 3.3 Cayley graph of the
C 3v point group. The
generators are c = ˆ
C 3 and
s =ˆ σ 1
A presentation of a group is a set of generators, together with a minimal set of relations that are sufficient to work out any product of two elements. As an example, let
us denote the ˆ
C 3 , ˆ
σ 1 generators as c, s. Just three relations among these generators
are sufficient to derive the whole multiplication table: c 3 = s 2 = e, sc = c 2 s.T h e
generation of the six elements of the group follows from Eq. (3.11):
ˆ
E = e
ˆ
C 3 = c
ˆ
C
2
3 = c
2
ˆ
σ 1 = s
ˆ
σ 2 = sc
ˆ
σ 3 = cs
(3.13)
Any product of these elements can be shown to be closed using only the presentation. As an example, the rule ˆ
σ 2 ˆ
σ 3 = ˆ
C 3 , is expressed in the presentation as follows:
sccs = s
c
2 s
= s(sc) = s
2 c = c
(3.14)
In this way the whole multiplication table can be derived.
The structure of the group can also be encoded in a graph known as the Cayley
graph. A graph is an abstract mathematical object consisting of a set of points, or
nodes, and a set of lines connecting pairs of these points. In a directed graph these
pairs are ordered, which means that directional arrows are added to the connecting
lines. In the Cayley graph every element of the group corresponds to a node. The
lines correspond to the action of the group generators. The generator ˆ
g connects
a given node ˆ
R i by a directed line to the resulting node ˆ g ˆ
R i . The action of
the group on its own Cayley graph will not only map nodes onto nodes, but will
also preserve the directed connections. As a result, the symmetry group will map
the graph onto itself. Such a mapping is called an automorphism. The group G is