Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Table 12.3.
1 i 1 i E A B C
1 1 i 1 i E E A B C
i i 1 i 1 or A A B C E
1 1 i 1 i B B C E A
i i 1 i 1 C C E A B
correspond respectively to the elements A 0 ; B 0 ; C 0 ; ... of G 0 , then the equation
AB ˆ C implies that A 0 B 0 ˆ C 0 , etc., and vice versa. Two isomorphic groups
have the same multiplication tables except for the labels attached to the group
elements. Obviously, two isomorphic groups must have the same order.
Groups that are isomorphic and so have the same multiplication table are the
same or identical, from an abstract point of view. That is why the concept of
isomorphism is a key concept to physicists. Diverse groups of operators that act
on diverse sets of objects have the same multiplication table; there is only one
abstract group. This is where the value and beauty of the group theoretical
method lie; the same abstract algebraic results may be applied in making predictions
about a wide variety physical objects.
The isomorphism of groups is a special instance of homomorphism, which
allows many-to-one correspondence.
Example 12.9
Consider the groups of Problems 12.2 and 12.4. The group G of Problem 12.2
consists of the four elements E ˆ 1; A ˆ i; B ˆ1; C ˆi with ordinary multiplication
as the rule of combination. The group multiplication table has the form
shown in Table 12.3. The group G 0 of Problem 12.4 consists of the following four
elements, with matrix multiplication as the rule of combination
E 0 ˆ 1 0
0 1
; A 0 ˆ
0
1
1 0
; B 0 ˆ 1 0
0 1
; C 0 ˆ 0 1
1 0
It is straightforward to check that the group multiplication table of group G 0 has
the form of Table 12.4. Comparing Tables 12.3 and 12.4 we can see that they have
precisely the same structure. The two groups are therefore isomorphic.
:
Example 12.10
We stated earlier that diverse groups of operators that act on diverse sets of
objects have the same multiplication table; there is only one abstract group. To
illustrate this, we consider, for simplicity, an abstract group of order two, G 2 : that
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