28.1 GROUPS
Using only the first equalities in (28.2) and (28.3), deduce the second ones.Consider the expressionX−^1 • (X•X−^1 );
X−^1 • (X•X−^1 )
(ii)
=(X−^1 • X)•X−^1(iv)
=I•X−^1
(iii)
=X−^1. (28.6)ButX−^1 belongs toG, and so from (iv) there is an elementUinGsuch that
U•X−^1 =I. (v)Form the product ofUwith the first and last expressions in (28.6) to give
U•(X−^1 • (X•X−^1 )) =U•X−^1
(v)
=I. (28.7)Transforming the left-hand side of this equation gives
U•(X−^1 • (X•X−^1 ))
(ii)
=(U•X−^1 )•(X•X−^1 )
(v)
=I•(X•X−^1 )
(iii)
=X•X−^1. (28.8)Comparing (28.7), (28.8) shows that
X•X−^1 =I, (iv)′i.e. the second equality in group definition (iv). Similarly
X•I
(iv)
=X•(X−^1 • X)(ii)
=(X•X−^1 )•X
(iv)′
=I•X
(iii)
=X. (iii′)i.e. the second equality in group definition (iii).
The uniqueness of the identity elementIcanalsobedemonstratedratherthanassumed. Suppose thatI′, belonging toG, also has the property
I′•X=X=X•I′ for allXbelonging toG.TakeXasI,then
I′•I=I. (28.9)Further, from (iii′),
X=X•I for allXbelonging toG,