In this chapter, we present the basic theory of finite groups and their representations as preparation for the discussion of continuous groups that starts from Chapter 3. It is assumed that readers know the basics of set theory, vector spaces, transformations, linear operators, matrix representations, inner products and such. These will be called upon as and when needed.

Definition of a Group

A group G is a set of elements a , b , c , · · · , g , g′ , · · · , e , · · · along with a composition (or ‘multiplication’) law obeying four conditions:

• (i) Closure: a , b ∈ G → ab = unique product element ∈ G .

• (ii) Associativity: for any a , b , c ∈ G ,

  a (bc ) = (ab )c = abc ∈ G .

• (iii) Identity: there is a unique element e ∈ G such that

  ae = ea = a , for any a ∈ G.

• (iv) Inverses: for each a ∈ G , there is a unique a−1 ∈ G , the inverse of a , such that

  a −1a = aa −1 = e . (1.1)

The composition rule or law can be called a binary law as the product is defined for any pair of elements. The conditions in (1.1) could be stated in more economical forms, for instance introducing only a left identity and left inverses, and then showing that the more general properties in (1.1) do hold.

One can immediately think of various qualitatively different possibilities. The number of (distinct) elements in G may be finite. Then this number, denoted by |G |, is called the order of G . Some other possibilities are that the number of elements may be a discrete infinity, or else a continuous infinity with G being a manifold of some dimension.

Some Examples

• (i) The symmetric group, the group of permutations on n objects, is finite, of order n !, and is denoted by Sn . We mention only a few pertinent properties now, and go into some more detail in Chapter 2.