Groups usually appear in physics as symmetries of the system or model we are studying. Often the symmetry operation involves a linear transformation, and this naturally leads to the idea of finding sets of matrices having the same multiplication table as the group. These sets are called representations of the group. Given a group, we endeavour to find and classify all possible representations.

Basic ideas

We begin with a rapid review of basic group theory.

Group axioms

A group G is a set with a binary operation that assigns to each ordered pair (g1, g2) of elements a third element, g3, usually written with multiplicative notation as g3 = g1g2. The binary operation, or product, obeys the following rules:

1. (i) Associativity: g1(g2g3) = (g1g2)g3.

2. (ii) Existence of an identity: there is an element e ∈ G such that eg = g for all g ∈ G.

3. (iii) Existence of an inverse: for each g ∈ G there is an element g–1 such that g–1g = e.

From these axioms there follow some conclusions that are so basic that they are often included in the axioms themselves, but since they are not independent, we state them as corollaries.