3.3 Some Special Groups
27
Table 3.3 Multiplication
table for the point group D 2
D 2
ˆ
E
ˆ
C
x
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
E
ˆ
E
ˆ
C x
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
C
x
2
ˆ
C
x
2
ˆ
E
ˆ
C
z
2
ˆ
C
y
2
ˆ
C
y
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
E
ˆ
C
x
2
ˆ
C
z
2
ˆ
C
z
2
ˆ
C
y
2
ˆ
C
x
2
ˆ
E
thus isomorphic to the automorphism group of its Cayley graph. The Cayley graph
corresponding to the group C 3v , generated by ˆ
C 3 and ˆ
σ 1 , is shown in Fig. 3.3.I t
resembles a trigonal prism, but with opposite directions in the upper and the lower
triangle. The ˆ
σ 1 generator corresponds to the upright edges of the prism. Since this
generator is its own inverse, these edges can be traversed in both directions, so they
are really undirected.
3.3 Some Special Groups
Abelian groups 1 are groups with a commutative multiplication rule, i.e.,
∀ ˆ
R ∈ G & ˆ
S ∈ G ⇒ ˆ
R ˆ
S = ˆ
S ˆ
R
(3.15)
Hence, in an abelian group, the multiplication table is symmetric about the diagonal.
Clearly, our group C 3v is not abelian.
Cyclic groups are groups with only one generator. They are usually denoted as
C n . The threefold axis gives rise to the cyclic group C 3 . Its elements consist of
products of the generator. By analogy with number theory, such multiple products
are called powers; hence, C 3 ={ ˆ
C 3 , ˆ
C 2
3 , ˆ
C 3
3 }, where the third power is of course
the unit element. Similarly, the reflection planes yield a cyclic group of order 2. The
standard notation for this group is not C 2 but C s . Cyclic groups are of course abelian
because the products of elements give rise to a sum of powers and summation is
commutative:
ˆ
C
i ˆ
C
j = ˆ
C
i+j = ˆ
C
j +i = ˆ
C
j ˆ
C
i
(3.16)
By contrast, not all abelian groups are cyclic. A simple example is the group 2 D 2 of
order 4, which is presented in Table 3.3. It needs two perpendicular twofold axes as
generators and thus cannot be cyclic. Nonetheless, it is abelian since its generators
commute.
The symmetric group, S n , is the group of all permutations of the elements of a
set of cardinality n. The order of S n is equal to n!. As it happens, our C 3v group
is isomorphic to S 3 . The permutations are defined on the ordered set of the three
1 Named after the Norwegian mathematician Niels Henrik Abel (1802–1829).
2 This group is isomorphic to Felix Klein’s four-group (Vierergruppe).
27
Table 3.3 Multiplication
table for the point group D 2
D 2
ˆ
E
ˆ
C
x
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
E
ˆ
E
ˆ
C x
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
C
x
2
ˆ
C
x
2
ˆ
E
ˆ
C
z
2
ˆ
C
y
2
ˆ
C
y
2
ˆ
C
y
2
ˆ
C
z
2
ˆ
E
ˆ
C
x
2
ˆ
C
z
2
ˆ
C
z
2
ˆ
C
y
2
ˆ
C
x
2
ˆ
E
thus isomorphic to the automorphism group of its Cayley graph. The Cayley graph
corresponding to the group C 3v , generated by ˆ
C 3 and ˆ
σ 1 , is shown in Fig. 3.3.I t
resembles a trigonal prism, but with opposite directions in the upper and the lower
triangle. The ˆ
σ 1 generator corresponds to the upright edges of the prism. Since this
generator is its own inverse, these edges can be traversed in both directions, so they
are really undirected.
3.3 Some Special Groups
Abelian groups 1 are groups with a commutative multiplication rule, i.e.,
∀ ˆ
R ∈ G & ˆ
S ∈ G ⇒ ˆ
R ˆ
S = ˆ
S ˆ
R
(3.15)
Hence, in an abelian group, the multiplication table is symmetric about the diagonal.
Clearly, our group C 3v is not abelian.
Cyclic groups are groups with only one generator. They are usually denoted as
C n . The threefold axis gives rise to the cyclic group C 3 . Its elements consist of
products of the generator. By analogy with number theory, such multiple products
are called powers; hence, C 3 ={ ˆ
C 3 , ˆ
C 2
3 , ˆ
C 3
3 }, where the third power is of course
the unit element. Similarly, the reflection planes yield a cyclic group of order 2. The
standard notation for this group is not C 2 but C s . Cyclic groups are of course abelian
because the products of elements give rise to a sum of powers and summation is
commutative:
ˆ
C
i ˆ
C
j = ˆ
C
i+j = ˆ
C
j +i = ˆ
C
j ˆ
C
i
(3.16)
By contrast, not all abelian groups are cyclic. A simple example is the group 2 D 2 of
order 4, which is presented in Table 3.3. It needs two perpendicular twofold axes as
generators and thus cannot be cyclic. Nonetheless, it is abelian since its generators
commute.
The symmetric group, S n , is the group of all permutations of the elements of a
set of cardinality n. The order of S n is equal to n!. As it happens, our C 3v group
is isomorphic to S 3 . The permutations are defined on the ordered set of the three
1 Named after the Norwegian mathematician Niels Henrik Abel (1802–1829).
2 This group is isomorphic to Felix Klein’s four-group (Vierergruppe).