Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
The same set of all integers does not form a group under multiplication. Why?
Because the inverses of integers are not integers and so they are not members of
the set.
Example 12.3
The set of all rational numbers (p=q, with q 6ˆ 0) forms a continuous in®nite group
under addition. It is an Abelian group, and we denote it by S 2 . The identity
element is 0; and the inverse of a given element is its negative.
Example 12.4
The set of all complex numbers …z ˆ x ‡ iy† forms an in®nite group under
addition. It is an Abelian group and we denote it by S 3 . The identity element
is 0; and the inverse of a given element is its negative (that is, z is the inverse
of z).
The set of elements in S 1 is a subset of elements in S 2 , and the set of elements in
S 2 is a subset of elements in S 3 . Furthermore, each of these sets forms a group
under addition, thus S 1 is a subgroup of S 2 ,andS 2 a subgroup of S 3 . Obviously
S 1 is also a subgroup of S 3 .
Example 12.5
The three matrices
~A ˆ 1 0
0 1
; ~B ˆ
0 1
1 1
; ~C ˆ
1 1
1 0
form an Abelian group of order three under matrix multiplication. The identity
element is the unit matrix, E ˆ ~A. The inverse of a given matrix is the inverse
matrix of the given matrix:
~A 1 ˆ 1 0
ˆ ~A; ~B 1 1 1
ˆ ˆ ~C; ~C 1 0 1
ˆ
ˆ ~B:
0 1
1 0
1 1
It is straightforward to check that all the four group axioms are satis®ed. We
leave this to the reader.
Example 12.6
The three permutation operations on three objects a; b; c
‰1 23Š; ‰2 31Š; ‰3 12Š
form an Abelian group of order three with sequential performance as the law of
combination.
The operation [1 2 3] means we put the object a ®rst, object b second, and object
c third. And two elements are multiplied by performing ®rst the operation on the
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