Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Obviously these two operators form a group that is isomorphic to G 2 . These two
groups (formed by the elements E 0 , R, and the elements ^O E 0 and ^O R , respectively)
are the two representations of the abstract group G 2 . These two simple examples
cannot illustrate the value and beauty of the group theoretical method, but they
do serve to illustrate the key concept of isomorphism.
Group of permutations and Cayley's theorem
In Example 12.6 we examined brie¯y the group of permutations of three objects.
We now come back to the general case of n objects (1; 2; ...; n) placed in n boxes
(or places) labeled 1 , 2 ; ...; n . This group, denoted by S n , is called the symmetric
group on n objects. It is of order n! How do we know? The ®rst object may
be put in any of n boxes, and the second object may then be put in any of n 1
boxes, and so forth: n…n 1†…n 2†3 2 1 ˆ n!:
We now de®ne, following common practice, a permutation symbol P
P ˆ 1 2 3 n
!
; …12:1†
1 2 3 n
which shifts the object in box 1 to box 1 , the object in box 2 to box 2 , and so
forth, where 1 2 n is some arrangement of the numbers 1; 2; 3; ...; n. The old
notation in Example 12.6 can now be written as
‰2 31Šˆ 1 2 3
:
2 3 1
For n objects there are n! permutations or arrangements, each of which may be
written in the form (12.1). Taking a speci®c example of three objects, we have
P 1 ˆ 1 2 3
; P 2 ˆ 1 2 3
; P 3 ˆ 1 2 3
;
1 2 3
2 3 1
1 3 2
P 4 ˆ 1 2 3
; P 5 ˆ 1 2 3
2 1 3
3 2 1
; P 6 ˆ 1 2 3
:
3 1 2
For the product of two permutations P i P j …i; j ˆ 1; 2; ...; 6†, we ®rst perform the
one on the right, P j , and then the one on the left, P i . Thus
P 3 P 6 ˆ 1 2 3
1 2 3
ˆ 1 2 3
ˆ P 4 :
1 3 2 3 1 2 2 1 3
To the reader who has diculty seeing this result, let us explain. Consider the ®rst
column. We ®rst perform P 6 , so that 1 is replaced by 3, we then perform P 3 and 3
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