chapter  1
Elements of Group Theory
Pages 30

This chapter introduces the concept of a group and presents the characteristics that are imperative for developing group theory as it applies to high energy physics.

What is a group? This can best be answered by reference to a few examples. Consider the set S {1,–1, i,–i}. The numbers in curly brackets are called its elements or members. With the law for the combination of elements, called the law of composition or the binary operation, as a multiplication of complex numbers, the elements of S possess the following four properties:

1. If an element of S is multiplied, according to the prescribed law of composition, by an element of the same set, the resulting number is again an element of S. This is called the closure property. For example,

1 t i i, i t (–i) 1, (–1) t (–1) 1

2. The multiplication is associative; that is, the product does not depend on the order in which the elements are multiplied. Thus, if a, b, and c are arbitrary elements of S, then a t (b t c) (a t b) t c. For example:

1 t {(–1) t i} 1 t (–i) –i


{1 t (–1)} t i (–1) t i –i

so that

1 t {(–1) t i} {1 t (–1)} t i

3. The set S contains an element that, when multiplied by any one of its elements, either from the left or from the right, reproduces that element. The element 1 in S possesses this characteristic:

1 t 1 1, 1 t (–1) –1, i t 1 i, 1 t (–i) –i

Such an element is said to be the identity element. 4. To every element of the set S, there corresponds an element of

the same set such that the product of two elements, irrespective of their order, is the identity element. Then any one of these elements is called the inverse of the other. For instance, multiplying the elements i and –i of S, we get 1 (i.e., the identity element). By virtue of the previous definition, –i is the inverse of i and i is the inverse of –i.