Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

ifHis anormalsubgroup ofGthen its (left) cosets themselves form a group (see

exercise 28.16).

28.7.3 Conjugates and classes

Our second example of an equivalence relation is concerned with those elements

XandYof a groupGthat can be connected by a transformation of the form

Y=G−i^1 XGi,whereGiis an (appropriate) element ofG. ThusX∼Yif there

exists an elementGiofGsuch thatY=G−i^1 XGi. Different pairs of elementsX

andYwill, in general, require different group elementsGi. Elements connected

in this way are said to beconjugates.

We first need to establish that this does indeed define an equivalence relation,

as follows.

(i) Reflexivity:X∼X,sinceX=I−^1 XIandIbelongs to the group.

(ii) Symmetry:X∼YimpliesY=G−i^1 XGiand thereforeX=(G−i^1 )−^1 YG−i^1.

SinceGibelongs toGso doesG−i^1 , and it follows thatY∼X.

(iii) Transitivity:X∼YandY∼ZimplyY=G−i^1 XGiandZ=G−j^1 YGj

and thereforeZ=G−j^1 G−i^1 XGiGj=(GiGj)−^1 X(GiGj). SinceGiandGj

belong toGso doesGiGj, from which it follows thatX∼Z.

These results establish conjugacy as an equivalence relation and hence show

that it dividesGinto classes, two elements being in the same class if, and only if,

they are conjugate.

Immediate corollaries are:

(i) IfZis in the class containingIthen

Z=G−i^1 IGi=G−i^1 Gi=I.

Thus, since any conjugate ofIcanbeshowntobeI, the identity must be

in a class by itself.

(ii) IfXis in a class by itself then

Y=G−i^1 XGi

must imply thatY=X.But

X=GiG−i^1 XGiG−i^1

for anyGi,andso

X=Gi(Gi−^1 XGi)G−i^1 =GiYG−i^1 =GiXG−i^1 ,

i.e.XGi=GiXfor allGi.

Thus commutation with all elements of the group is a necessary (and

sufficient) condition for any particular group element to be in a class by

itself. In an Abelian group each element is in a class by itself.