60 I The Expanding Universe of Numbers
subgroupgeneratedbyS. ClearlyS⊆〈S〉and〈S〉is contained in every subgroup of
Gwhich containsS.
Two e l e m e n t sa,bof a groupGare said to beconjugateifb=x−^1 axfor some
x ∈G. It is easy to see that conjugacy is an equivalence relation. Fora=a−^1 aa,
ifb=x−^1 axthena=(x−^1 )−^1 bx−^1 ,andb=x−^1 ax,c=y−^1 bytogether imply
c=(xy)−^1 axy. ConsequentlyGmay be partitioned intoconjugacy classes,sothat
two elements ofGare conjugate if and only if they belong to the same conjugacy class.
For any elementaof a groupG,thesetNaof all elements ofGwhich commute
witha,
Na={x∈G:xa=ax},is closed under multiplication and inversion. ThusNais a subgroup ofG, called the
centralizerofainG.
Ifyandzlie in the same right coset ofNa,sothatz=xyfor somex∈Na,then
zy−^1 a=azy−^1 and hencey−^1 ay=z−^1 az.Conversely,ify−^1 ay=z−^1 az,thenyand
zlie in the same right coset ofNa.IfGis finite, it follows that the number of elements
in the conjugacy class containingais equal to the number of right cosets of the sub-
groupNa,i.e.totheindexof the subgroupNainG, and hence it divides the order ofG.
To conclude, we mention a simple way of creating new groups from given ones.
LetG,G′be groups and letG×G′be the set of all ordered pairs (a,a′) witha∈Gand
a′∈G′.ThenG×G′acquires the structure of a group if we define the product(a,a′)·
(b,b′)of(a,a′)and(b,b′)to be(ab,a′b′). Multiplication is clearly associative,(e,e′)
is an identity element and(a−^1 ,a′−^1 )is an inverse for (a,a′). The group thus con-
structed is called thedirect productofGandG′, and is again denoted byG×G′.
8 RingsandFields
A nonempty setR is said to be aringif two binary operations,+(addition)
and·(multiplication), are defined with the properties
(i) Ris a commutative group under addition, with 0 (zero) as identity element and
−aas inverse ofa;
(ii) multiplication is associative:(ab)c=a(bc)for alla,b,c∈R;
(iii) there exists an identity element 1 (one) for multiplication:a 1 =a= 1 afor every
a∈R;
(iv) addition and multiplication are connected by the two distributive laws:
(a+b)c=(ac)+(bc), c(a+b)=(ca)+(cb) for alla,b,c∈R.The elements 0 and 1 are necessarily uniquely determined. If, in addition, multi-
plication is commutative:
ab=ba for alla,b∈R,thenRis said to be acommutativering. In a commutative ring either one of the two
distributive laws implies the other.