Application Of Group Theory To Physics

ABSTRACT

Let G be a group and F be a representative of the group.

In this thesis, I intend to study the application of group theory to physics.

Some Basic Concept of Group Theory

Definition: A non - empty set G is called a group with respect to an

operation defined on G if following conditions are satisfied:

(i) Closure Property: a * b E G for all a, b E G.

(ii) Associative Property: (a * b)*c = a (b * c) for all a, b, c, E G

(iii) Identify Element: There exists an element e E G such that

aee*a=aforalla E G

(iv) Inverse Element: For every a E G, there exists a' E G

suchthata*a'a' a=e

Definition: A group G is said to be Abelian or commutative if a * b = b * a for

all a, b E G

Definition: The number of elements in a group is called the order of the group

and is denoted by o (G) or IG. If G is finite, then o (G) = n otherwise

o(G)=oo

Some Examples of Groups: Groups abound in most branches of mathematics.

Here, we assemble a few examples:

(i) The set G ={ 1, - 1,i,-i} is an Abelian group under multiplication and o (G) 4.

iii) The set G = { ( )\a E R* } is a group under matrix

multiplication and o(G)

(iii) The set of all integers forms an Abelian group with respect to addition.

, (iv) The set of all positive rational numbers forms a group with respect to

multiplication.