58 I The Expanding Universe of Numbers
The relation is certainly reflexive, sincee∈H. It is also symmetric, since ifc=
ba−^1 ∈H,thenc−^1 =ab−^1 ∈H. Furthermore it is transitive, since ifba−^1 ∈Hand
cb−^1 ∈H,thenalsoca−^1 =(cb−^1 )(ba−^1 )∈H.
The equivalence class which containsais the setHaof all elementsha,where
h∈H. We call any such equivalence class aright cosetof the subgroupH,andany
element of a given coset is said to be arepresentativeof that coset.
It follows from the remarks in§0 about arbitrary equivalence relations that, for any
two cosetsHaandHa′, eitherHa=Ha′orHa∩Ha′=∅. Moreover, the distinct
right cosets form a partition ofG.
IfHis a subgroup of a finite groupG,thenHis also finite and the number of
distinct right cosets is finite. Moreover,each right coset Ha contains the same number
of elements asH, since the mappingh→haofHtoHais bijective. It follows that
the order of the subgroupHdivides the order of the whole groupG, a result usually
known asLagrange’s theorem. The quotient of the orders, i.e. the number of distinct
cosets, is called theindexofHinG.
Suppose again thatHis a subgroup of an arbitrary groupGand thata,b∈G.By
writinga∼lbifa−^1 b∈H, we obtain another equivalence relation. The equivalence
class which containsais now the setaHof all elementsah,whereh∈H. We call
any such equivalence class aleft cosetof the subgroupH. Again, two left cosets either
coincide or are disjoint, and the distinct left cosets form a partition ofG.
When are the two partitions, into left cosets and into right cosets, the same? Evi-
dentlyHa=aHfor everya∈Gif and only ifa−^1 Ha=Hfor everya∈Gor,
sinceamay be replaced bya−^1 , if and only ifa−^1 ha∈Hfor everyh∈Hand every
a∈G. A subgroupHwhich satisfies this condition is said to be ‘invariant’ ornormal.
Any groupGobviously has two normal subgroups, namelyGitself and the subset
{e}which contains only the identity element. A groupGis said to besimpleif it has
no other normal subgroups and if these two are distinct (i.e.,Gcontains more than one
element).
We now show that ifHis a normal subgroup of a groupG, then the collection of
all cosets ofHcan be given the structure of a group. SinceHa=aHandHH=H,
we have
(Ha)(Hb)=H(Ha)b=Hab.Thus if we define the productHa·Hbof the cosetsHaandHbto be the cosetHab,
the definition does not depend on the choice of coset representatives. Clearly multipli-
cation of cosets is associative, the cosetH=Heis an identity element and the coset
Ha−^1 is an inverse of the cosetHa. The new group thus constructed is called thefactor
grouporquotient groupofGby the normal subgroupH, and is denoted byG/H.
A mapping f:G →G′of a groupGinto a groupG′is said to be a (group)
homomorphismif
f(ab)=f(a)f(b) for alla,b∈G.By takinga=b=e, we see that this implies thatf(e)=e′is the identity element
ofG′.Bytakingb=a−^1 , it now follows that f(a−^1 )is the inverse of f(a)inG′.
Since the subsetf(G)ofG′is closed under both multiplication and inversion, it is a
subgroup ofG′.