The general theory of groups and group representations for finite groups was outlined in Chapter 1. This theory was utilised in Chapter 2 to describe some features of symmetric permutation groups. Here, we will move on to continuous groups, beginning with the results from the finite case and generalising them in a heuristic manner. For continuous groups we go beyond algebraic methods and use calculus very effectively.

As our first examples we look at rotations in two and three dimensions which are important for physics. When needed we will call upon results from the quantum theory of angular momentum. We will look at definitions, structures, topologies and representations mainly in a descriptive manner.

The Group SO(2)

This is the group of proper rotations in a plane, with respect to a chosen origin, with the defining matrix representation

SO (2) = ﹛A = 2 × 2 real matrix|AT A = 11, det A = 1﹜

This is a one parameter or one dimensional continuous abelian group. The group space or manifold is the circle 𝕤1. The composition law, identity and inverses are:

all arguments taken modulo 2π . As this is an abelian group, each element is a class by itself, the centre is the whole group, and the commutator subgroup is trivial.

For every positive integer N , the set of elements ﹛A ( 2πn/N) , n = 0, 1, · · · ,N − 1﹜ is a discrete subgroup of order N.

The topology of SO (2) is nontrivial. While it is certainly connected – one can move continuously from any element to any other without leaving the group manifold – it has infinite connectivity. There is one class of continuous closed loops starting and ending at the identity for each ‘winding number’ n = 0,±1,±2, · · ·.