526 Chapter 18Matrices and linear transformations
Other symmetry operations (reflections) are possible, but are equivalent to the
above six operations for the planefigure. The operations may also be interpreted as
permutations of the labels (1, 2, 3) of the three points.
The successive application of any two symmetry operations is equivalent to
the application of a single operation. The two examples in Figure 18.8 show that the
application of operation Afollowed by Cis equivalent to the application of the single
operation D, and that Cfollowed by Ais equivalent to F. Such combinations of
symmetry operations are denoted by the symbolic equations
CA 1 = 1 D, AC 1 = 1 F (18.66)
The results of the possible combinations of pairs of operations are collected in the
group multiplication table18.1.
Table 18.1 A group multiplication table
EAB CD F(applied first)
EEAB CD F
AAB E F C D
BBE ADF C
CCDF E A B
DDF C B E A
FF C D A B E
The six operations form a closed set called a group; we will refer to this group as
G 1 = 1 {E,A, B, C, D, F}. A symmetry group that is made up of rotations, reflections,
and inversion is called a point groupbecause at least one point is left unmoved by
every symmetry operation. There is only one such point in our example, the centroid
of the triangle.
0 Exercise 62
............................................................................
...
...
...
..
...
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
...
................. .........................................................1
23
••
............................................................................
.........
................................................................................................
.........
....................CA
...............................................
..
...
...
..
...
...
..
...
...
..
...
..F
............................................................................
...
...
...
..
...
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
...
................. .........................................................1
32
••
............................................................................
...
...
...
..
...
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
...
................. .........................................................2
13
••
...........................................................................
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
....
..
...
...
...
...
...
................. .........................................................1
23
••
...........................................................................
........
......................
...........................................................................
........
......................AC
.................................................
..
...
..
...
......
..
...
..
...
...D
...........................................................................
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
....
..
...
...
...
...
...
................. .........................................................3
12
••
...........................................................................
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
....
..
...
...
...
...
...
................. .........................................................3
21
••
Figure 18.8