NonAbelian groups

GROUP THEORY
1
i
−1
−i
1
1
i
−1
−i
i
i
−1
−i
1
−1
−1
−i
1
i
−i
−i
1
i
−1
1
2
4
3
1
1
2
4
3
2
2
4
3
1
4
4
3
1
2
3
3
1
2
4
Table 28.5 A comparison between tables 28.4 and 28.2(b), the latter with its
columns reordered.
I
A
B
C
I
I
A
B
C
A
A
B
C
I
B
B
C
I
A
C
C
I
A
B
Table 28.6 The common structure exemplified by tables 28.4 and 28.2(b), the
latter with its columns reordered.
amount of relabelling (or, equivalently, no allocation of the symbols A, B, C, amongst
i, −1, −i) can bring table 28.4 into the form of table 28.3. We conclude that the group
{1, i, −1, −i} is not isomorphic to S or S . An alternative way of stating the observation is
to say that the group contains only one element of order 2 whilst a group corresponding
to table 28.3 contains three such elements.
However, if the rows and columns of table 28.2(b) – in which the identity does appear
twice on the diagonal and which therefore has the potential to be equivalent to table 28.4 –
are rearranged by making the heading order 1, 2, 4, 3 then the two tables can be compared
in the forms shown in table 28.5. They can thus be seen to have the same structure, namely
that shown in table 28.6.
We therefore conclude that the group of four elements {1, i, −1, −i} under ordinary multiplication of complex numbers is isomorphic to the group {1, 2, 3, 4} under multiplication
(mod 5). What we have done does not prove it, but the two tables 28.3 and 28.6 are in
fact the only possible tables for a group of order 4, i.e. a group containing exactly
four elements.
28.3 Non-Abelian groups
So far, all the groups for which we have constructed multiplication tables have
been based on some form of arithmetic multiplication, a commutative operation,
with the result that the groups have been Abelian and the tables symmetric
about the leading diagonal. We now turn to examples of groups in which some
non-commutation occurs. It should be noted, in passing, that non-commutation
cannot occur throughout a group, as the identity always commutes with any
element in its group.
1052
28.3 NON-ABELIAN GROUPS
As a first example we consider again as elements of a group the two-dimensional
operations which transform an equilateral triangle into itself (see the end of
subsection 28.1.1). It has already been shown that there are six such operations:
the null operation, two rotations (by 2π/3 and 4π/3 about an axis perpendicular
to the plane of the triangle) and three reflections in the perpendicular bisectors
of the three sides. To abbreviate we will denote these operations by symbols as
follows.
(i) I is the null operation.
(ii) R and R are (clockwise) rotations by 2π/3 and 4π/3 respectively.
(iii) K, L, M are reflections in the three lines indicated in figure 28.2.
Some products of the operations of the form X • Y (where it will be recalled
that the symbol • means that the second operation X is carried out on the system
resulting from the application of the first operation Y ) are easily calculated:
R • R = R,
R • R = R,
R • R = I = R • R
(28.12)
K • K = L • L = M • M = I.
Others, such as K • M, are more difficult, but can be found by a little thought,
or by making a model triangle or drawing a sequence of diagrams such as those
following.
x
K •M
x
= K
=
=
R
x
x
showing that K • M = R . In the same way,
M•K
=
= M
x
x
=
R
x
x
shows that M • K = R, and
R•L
=
= R
x
x
=
x
K
x
shows that R • L = K.
Proceeding in this way we can build up the complete multiplication table
(table 28.7). In fact, it is not necessary to draw any more diagrams, as all
remaining products can be deduced algebraically from the three found above and
1053
GROUP THEORY
I
R
R
K
L
M
I
I
R
R
K
L
M
R
R
R
I
L
M
K
R
R
I
R
M
K
L
K
K
M
L
I
R
R
L
L
K
M
R
I
R
M
M
L
K
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 is not symmetric 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 if I is 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 nonAbelian 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 set M of six orthogonal 2 × 2 matrices given by
√
√ 3
− 12
− 12 −2 3
1 0
2
√
√
B=
I=
A=
3
0 1
− 21
− 23 − 12
2
(28.13)
√
√ 1
1
3
3
−
−1 0
2
2
2
2
√
√
E=
C=
D=
3
0 1
− 3 −1
−1
2
2
2
2
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 any
two 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 matrix I clearly 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, D and E being their own inverses. The group table is shown in table 28.8.
1054
28.3 NON-ABELIAN GROUPS
I
I
A
B
C
D
E
I
A
B
C
D
E
A
A
B
I
D
E
C
B
B
I
A
E
C
D
C
C
E
D
I
B
A
D
D
C
E
A
I
B
E
E
D
C
B
A
I
Table 28.8 The group table, under matrix multiplication, for the set M of
six orthogonal 2 × 2 matrices given by (28.13).
The similarity to table 28.7 is striking. If {R, R , K, L, M} of that table are
replaced by {A, B, C, D, E} respectively, the two tables are identical, without even
the need to reshuffle the rows and columns. The two groups, one of reflections
and rotations of an equilateral triangle, the other of matrices, are isomorphic.
Our second example of a group isomorphic to the same rotation–reflection
group is provided by a set of functions of an undetermined variable x. The
functions are as follows:
f1 (x) = x,
f2 (x) = 1/(1 − x),
f3 (x) = (x − 1)/x,
f4 (x) = 1/x,
f5 (x) = 1 − x,
f6 (x) = x/(x − 1),
and the law of combination is
fi (x) • fj (x) = fi (fj (x)),
i.e. the function on the right acts as the argument of the function on the left to
produce a new function of x. It should be emphasised that it is the functions
that are the elements of the group. The variable x is the ‘system’ on which they
act, and plays much the same role as the triangle does in our first example of a
non-Abelian group.
To show an explicit example, we calculate the product f6 • f3 . The product
will be the function of x obtained by evaluating y/(y − 1), when y is set equal to
(x − 1)/x. Explicitly
f6 (f3 ) =
(x − 1)/x
= 1 − x = f5 (x).
(x − 1)/x − 1
Thus f6 • f3 = f5 . Further examples are
f2 • f2 =
x−1
1
=
= f3 ,
1 − 1/(1 − x)
x
f6 • f6 =
x/(x − 1)
= x = f1 .
x/(x − 1) − 1
and
1055
(28.14)