Chemistry Reference
In-Depth Information
Ta b l e 3 . 3
Multiplication
table for the point group
D
2
E
C
2
C
2
C
2
D
2
E
E
C
2
C
2
C
2
C
2
C
2
C
2
E
C
2
C
2
C
2
C
2
E
C
2
C
2
C
2
C
2
C
2
E
thus isomorphic to the automorphism group of its Cayley graph. The Cayley graph
corresponding to the group
C
3
v
, generated by
C
3
and
σ
1
, is shown in Fig.
3.3
.It
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.
ˆ
3.3 Some Special Groups
Abelian
groups
1
are groups with a commutative multiplication rule, i.e.,
∀
R
S
⇒
RS
=
S R
∈
G
&
∈
G
(3.15)
Hence, in an abelian group, the multiplication table is symmetric about the diagonal.
Clearly, our group
C
3
v
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
={
C
3
, C
3
, C
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:
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
!
C
i
C
j
=
C
i
+
j
=
C
j
+
i
=
C
j
. As it happens, our
C
3
v
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
).
