18.7 Symmetry operations 527
Axioms of group theory
A set of elements{E, P, Q, R, =}forms a group if the following conditions are
satisfied.
(i)The combination of any pair of the elements of the group also belongs to the
group.
The law of combination depends on the nature of the elements; for example, addition
or multiplication if the elements are numbers, matrix multiplication if they are
matrices, consecutive application of symmetry or other operations. The combination
of two elements,PandQ, is called the product of Pand Qand is written asPQ,
with some convention about the ordering of the elements. The associative law of
combination must hold for all the elements of the group;P(QR) 1 = 1 (PQ)R 1 = 1 PQR. The
commutative law does not necessarily hold;PQ 1 ≠ 1 QPin general; for example, for the
group G,AC 1 ≠ 1 CA. IfPQ 1 = 1 QPfor all the elements of the group, the group is called
an Abelian group.
(ii)One of the elements of the group, denoted by E, has the properties of a unit (or
identity) element. For every element P,
PE 1 = 1 EP 1 = 1 P
(iii)Each element has an inverse which also belongs to the group: if Pbelongs to the
group then its inverseP
− 1is defined by
PP
− 11 = 1 P
− 1P 1 = 1 E
For the groupG 1 = 1 {E, A, B, C, D, F}the inverse elements are
E
− 11 = 1 E, A
− 11 = 1 B, B
− 11 = 1 A, C
− 11 = 1 C, D
− 11 = 1 D, F
− 11 = 1 F
Matrix representations of groups
A set of matrices that multiply in accordance with the multiplication table of a group
is called a matrix representationΓof the group. Three representations of the group
G 1 = 1 {E, A, B, C, D, F}are shown in Table 18.2.
Table 18.2 Matrix representations of the group G
G EA B CD F
Γ
1
11 1 11 1
Γ
2
11 1 − 1 − 1 − 1
Γ
3
The representationsΓ
1andΓ
2are called one-dimensional representations (the matrices
are the numbers± 1 ). The representationΓ
1, in which every symmetry operation is
1
2
3
2
3
2
1
2
−
−−
1
2
3
2
3
2
1
2
−
−
10
01
−
−−
1
2
3
2
3
2
1
2
−−
−
1
2
3
2
3
2
1
2
10
01