28

Group theory

For systems that have some degree of symmetry, full exploitation of that symmetry

is desirable. Significant physical results can sometimes be deduced simply by a

study of the symmetry properties of the system under investigation. Consequently

it becomes important, for such a system, to identify all those operations (rotations,

reflections, inversions) that carry the system into a physically indistinguishable

copy of itself.

The study of the properties of the complete set of such operations forms

one application ofgroup theory. Though this is the aspect of most interest to

the physical scientist, group theory itself is a much larger subject and of great

importance in its own right. Consequently we leave until the next chapter any

direct applications of group theoretical results and concentrate on building up

the general mathematical properties of groups.

28.1 Groups

As an example of symmetry properties, let us consider the sets of operations,

such as rotations, reflections, and inversions, that transform physical objects, for

example molecules, into physically indistinguishable copies of themselves, so that

only the labelling of identical components of the system (the atoms) changes in

the process. For differently shaped molecules there are different sets of operations,

but in each case it is a well-defined set, and with a little practice all members of

each set can be identified.

As simple examples, consider (a) the hydrogen molecule, and (b) the ammonia

molecule illustrated in figure 28.1. The hydrogen molecule consists of two atoms

H of hydrogen and is carried into itself by any of the following operations:

(i) any rotation about its long axis;

(ii) rotation throughπabout an axis perpendicular to the long axis and

passing through the pointMthat lies midway between the atoms;