Number Theory: An Introduction to Mathematics

7 Groups 55

Consequently the norm is multiplicative:for allα,β∈O,

n(αβ)=n(α)n(β).

For, puttingγ=αβ,wehave

n(γ)α ̄=(αγ) ̄ γ ̄=(α(αβ)) ̄ γ ̄=n(α)βγ ̄=n(α)β(β ̄α) ̄ =n(α)n(β)α. ̄

This establishes the result whenα=0, and whenα=0 it is obvious.
Everyα∈Ohas a unique representationα=a 1 +a 2 ε,wherea 1 ,a 2 ∈H,and
hence a unique representation

α=c 0 +c 1 i+c 2 j+c 3 k+c 4 ε+c 5 iε+c 6 jε+c 7 kε,

wherec 0 ,...,c 7 ∈R.Sinceα ̄=a 1 −a 2 εandn(α)=a 1 a 1 +a 2 a 2 , it follows that

α ̄=c 0 −c 1 i−c 2 j−c 3 k−c 4 ε−c 5 iε−c 6 jε−c 7 kε

and
n(α)=c^20 +···+c^27.

Consequently the relationn(α)n(β)=n(αβ)may be written in the form

(c^20 +···+c^27 )(d 02 +···+d 72 )=e 02 +···+e 72 ,

whereei=

∑ 7

j= 0

∑ 7

k= 0 ρijkcjdkfor some real constantsρijkwhich do not depend on
thec’s andd’s. An ‘8-squares identity’ of this type was first found by Degen (1818).

7 Groups

A nonempty setGis said to be agroupif a binary operationφ, i.e. a mapping
φ:G×G→G, is defined with the properties

(i)φ(φ(a,b),c)=φ(a,φ(b,c))for alla,b,c∈G; (associative law)

(ii) there existse∈Gsuch thatφ(e,a)=afor everya∈G; (identity element)

(iii) for eacha∈G, there existsa−^1 ∈Gsuch thatφ(a−^1 ,a)=e.(inverse elements)

If, in addition,

(iv)φ(a,b)=φ(b,a)for alla,b∈G,(commutative law)

then the groupGis said to becommutativeorabelian.
For example, the setZof all integers is a commutative group under addition, i.e.
withφ(a,b)=a+b, with 0 as identity element and−aas the inverse ofa. Similarly,
the setQ×of all nonzero rational numbers is a commutative group under multiplica-
tion, i.e. withφ(a,b)=ab, with 1 as identity element anda−^1 as the inverse ofa.
We now give an example of a noncommutative group. The setSAof all bijective
mapsf:A→Aof a nonempty setAto itself is a group under composition, i.e. with
φ(a,b)=a◦b, with the identity mapiAas identity element and the inverse mapf−^1
as the inverse off.IfAcontains at least 3 elements, thenSAis a noncommutative