GROUP THEORY
IRR′ KLM
I IRR′ KLM
R RR′ IMKL
R′ R′ IRLMK
K KLMI RR′
L LMKR′ IR
M MK L R R′ I
Table 28.7 The group table for the two-dimensional symmetry operations on
an equilateral triangle.the more self-evident results given in (28.12). A number of things may be noticed
about this table.
(i) It isnotsymmetric about the leading diagonal, indicating that some pairs
of elements in the group do not commute.
(ii) There is some symmetry within the 3×3 blocks that form the four quarters
of the table. This occurs because we have elected to put similar operations
close to each other when choosing the order of table headings – the two
rotations (or three ifIis viewed as a rotation by 0π/3) are next to each
other, and the three reflections also occupy adjacent columns and rows.
We will return to this later.That two groups of the same order may be isomorphic carries over to non-Abelian groups. The next two examples are each concerned with sets of six
objects; they will be shown to form groups that, although very different in nature
from the rotation–reflection group just considered, are isomorphic to it.
We consider first the setMof six orthogonal 2×2 matrices given byI=(
10
01)
A=(
−^12√
3
2
−√
3
2 −1
2)B=(
−^12 −√
3
√^2
3
2 −1
2)C=(
− 10
01)
D=( 1
2 −√
3
2
−√
3
2 −1
2)E=( 1
2√
3
√^2
3
2 −1
2)(28.13)the combination law being that of ordinary matrix multiplication. Here we use
italic, rather than the sans serif used for matrices elsewhere, to emphasise that
the matrices are group elements.
Although it is tedious to do so, it can be checked that the product of anytwo of these matrices, in either order, is also in the set. However, the result is
generally different in the two cases, as matrix multiplication is non-commutative.
The matrixIclearly acts as the identity element of the set, and during the checking
for closure it is found that the inverse of each matrix is contained in the set,I,
C,DandEbeing their own inverses. The group table is shown in table 28.8.