Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.1 GROUPS

28.1.1 Definition of a group

AgroupGis a set of elements{X,Y,...}, together with a rule for combining

them that associates with each ordered pairX,Ya ‘product’ or combination law

X•Yfor which the following conditions must be satisfied.

(i) Foreverypair of elementsX, Ythat belongs toG, the productX•Yalso

belongs toG. (This is known as theclosure propertyof the group.)

(ii) For all triplesX, Y , Ztheassociative lawholds; in symbols,

X•(Y•Z)=(X•Y)•Z. (28.1)

(iii) There exists a unique elementI, belonging toG, with the property that

I•X=X=X•I (28.2)

forallXbelonging toG. This elementIis known as theidentity element

of the group.

(iv) For every elementXofG, there exists an elementX−^1 , also belonging to

G, such that

X−^1 • X=I=X•X−^1. (28.3)

X−^1 is called theinverseofX.

An alternative notation in common use is to write the elements of a groupG

as the set{G 1 ,G 2 ,...}or, more briefly, as{Gi}, a typical element being denoted

byGi.

It should be noticed that, as given, the nature of the operation•is not stated. It

should also be noticed that the more general termelement, rather thanoperation,

has been used in this definition. We will see that the general definition of a

group allows as elements not only sets of operations on an object but also sets of

numbers, of functions and of other objects, provided that the interpretation of•

is appropriately defined.

In one of the simplest examples of a group, namely the group of all integers

under addition, the operation•is taken to be ordinary addition. In this group the

role of the identityIis played by the integer 0, and the inverse of an integerXis

−X. That requirements (i) and (ii) are satisfied by the integers under addition is

trivially obvious. A second simple group, under ordinary multiplication, is formed

by the two numbers 1 and−1; in this group, closure is obvious, 1 is the identity

element, and each element is its own inverse.

It will be apparent from these two examples that the number of elements in a

group can be either finite or infinite. In the former case the group is called afinite

groupand the number of elements it contains is called theorderof the group,

which we will denote byg; an alternative notation is|G|but has obvious dangers