Chemistry Reference
In-Depth Information
has very remarkable properties. Each row and each column represent a permutation
of the ordered set of elements, but in such a way that every element occurs only
once in each row and column. This is a direct consequence of the group properties.
As in many group-theoretical proofs, the simplest way to show this is by a
reductio
ad absurdum
. Suppose that a given element,
T
, occurred at entries
ij
and
ik
, with
R
k
=
R
j
. Then one would have, by applying the rules:
R
i
R
j
=
R
i
R
k
R
−
1
i
( R
i
R
j
)
=
R
−
1
i
( R
i
R
k
)
R
−
1
i
R
i
R
j
=
R
−
1
R
i
R
k
(3.9)
i
E R
j
=
E R
k
R
j
=
R
k
which would contradict the original supposition. Along the same lines it is easy to
prove that the inverse of a product is equal to the product of the inverses in the
opposite order:
( R
i
R
j
)
−
1
=
R
−
1
j
R
−
1
i
(3.10)
As a matter of principle, the group multiplication table contains everything there is
to know about the group. It is, though, not necessary to store the whole multiplica-
tion table. A more compact way uses
generators
. The generators are defined as a
minimal set of elements capable of generating the whole group. For the present ex-
ample, two generators are needed, e.g.,
C
3
and
σ
1
. It is sufficient to make all binary
combinations of these two operators in order to generate all remaining elements:
C
3
C
3
=
C
3
ˆ
σ
1
=
E
C
3
ˆ
σ
1
ˆ
(3.11)
σ
1
=ˆ
σ
3
σ
1
C
3
=ˆ
ˆ
σ
2
ˆ
Alternatively, any pair of reflection planes would suffice as generators, say
σ
1
and
σ
2
, but in this case the remaining symmetry plane can be obtained only by a
further multiplication:
σ
1
=
E
σ
1
ˆ
ˆ
σ
2
=
E
σ
2
ˆ
ˆ
σ
2
=
C
3
(3.12)
σ
1
ˆ
ˆ
σ
1
=
C
3
σ
2
ˆ
ˆ
σ
1
ˆ
ˆ
σ
2
ˆ
σ
1
=ˆ
σ
3
