Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

and settingX=I′gives

I′=I′•I. (28.10)

It then follows from (28.9), (28.10) thatI=I′, showing that in any particular

group the identity element is unique.

In a similar way it can be shown that the inverse of any particular element

is unique. IfUandVare two postulated inverses of an elementXofG,by

considering the product

U•(X•V)=(U•X)•V,

it can be shown thatU=V. The proof is left to the reader.

Given the uniqueness of the inverse of any particular group element, it follows

that

(U•V•···•Y•Z)•(Z−^1 • Y−^1 • ···•V−^1 • U−^1 )

=(U•V•···•Y)•(Z•Z−^1 )•(Y−^1 • ···•V−^1 • U−^1 )

=(U•V•···•Y)•(Y−^1 • ···•V−^1 • U−^1 )

..

.

=I,

where use has been made of the associativity and of the two equationsZ•Z−^1 =I

andI•X=X. Thus the inverse of a product is the product of the inverses in

reverse order, i.e.

(U•V•···•Y•Z)−^1 =(Z−^1 • Y−^1 • ···•V−^1 • U−^1 ). (28.11)

Further elementary results that can be obtained by arguments similar to those

above are as follows.

(i) Given any pair of elementsX, Y belonging toG, there exist unique

elementsU, V, also belonging toG, such that

X•U=Y and V•X=Y.

ClearlyU=X−^1 • Y,andV=Y•X−^1 , and they can be shown to be

unique. This result is sometimes called thedivision axiom.

(ii) Thecancellation lawcan be stated as follows. If

X•Y=X•Z

for someXbelonging toG,thenY=Z.Similarly,

Y•X=Z•X

implies the same conclusion.