Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

Three immediate consequences of the above definition are proved as follows.

(i) IfIis the identity ofGthenIX=Xfor allXinG.Consequently

X′=(IX)′=I′X′,

for allX′inG′. ThusI′is the identity inG′. In words, the identity element

ofGmaps into the identity element ofG′.

(ii) Further,

I′=(XX−^1 )′=X′(X−^1 )′.

That is, (X−^1 )′=(X′)−^1 .In words, the image of an inverse is the same

element inG′as the inverse of the image.

(iii) If elementXinGis of orderm,i.e.I=Xm,then

I′=(Xm)′=(XXm−^1 )′=X′(Xm−^1 )′=···=X︸′X′︷︷···X︸′

mfactors

.

In words, the image of an element has the same order as the element.

What distinguishes an isomorphism from the more general homomorphism are

the requirements that in an isomorphism:

(I) different elements inGmust map into different elements inG′(whereas in

a homomorphism several elements inGmay have the same image inG′),

that is,x′=y′must implyx=y;

(II) any element inG′must be the image of some element inG.

An immediate consequence of (I) and result (iii) for homomorphisms is that

isomorphic groups each have the same number of elements of any given order.

For a general homomorphism, the set of elements ofGwhose image inG′

isI′is called thekernelof the homomorphism; this is discussed further in the

next section. In an isomorphism the kernel consists of the identityIalone. To

illustrate both this point and the general notion of a homomorphism, consider

a mapping between the additive group of real numbers and the multiplicative

group of complex numbers with unit modulus,U(1). Suppose that the mapping

→U(1) is

Φ:x→eix;

then this is a homomorphism since

(x+y)′→ei(x+y)=eixeiy=x′y′.

However, it is not an isomorphism because many (an infinite number) of the

elements of have the same image inU(1). For example,π, 3 π, 5 π,...in all

have the image−1inU(1) and, furthermore, all elements of of the form 2πn,

wherenis an integer, map onto the identity element inU(1). The latter set forms

the kernel of the homomorphism.