7 Groups 59Ifg:G′→G′′is a homomorphism of the groupG′into a groupG′′, then the
composite mapg◦f:G→G′′is also a homomorphism.
Thekernelof the homomorphismfis defined to be the setNof alla∈Gsuch
thatf(a)=e′is the identity element ofG′. The kernel is a subgroup ofG, since if
a∈Nandb∈N,thenab∈Nanda−^1 ∈N. Moreover, it is a normal subgroup,
sincea∈Nandc∈Gimplyc−^1 ac∈N.
For anya∈G, puta′= f(a)∈G′.ThecosetNais the set of allx∈Gsuch
thatf(x)=a′,andthemapNa→a′is a bijection from the collection of all cosets
ofNtof(G).Sincefis a homomorphism,Nabis mapped toa′b′. Hence the map
Na→a′is a homomorphism of the factor groupG/Ntof(G).
A mappingf:G→G′of a groupGinto a groupG′is said to be a (group)isomor-
phismif it is both bijective and a homomorphism. The inverse mappingf−^1 :G′→G
is then also an isomorphism. (Anautomorphismof a groupGis an isomorphism ofG
with itself.)
Thus we have shown that, iff:G→G′is a homomorphism of a groupGinto a
groupG′, with kernelN, then the factor groupG/Nis isomorphic tof(G).
Suppose now thatGis an arbitrary group andaany element ofG.Wehave
already defineda−^1 ,theinverseofa. We now inductively definean, for any integern,
by putting
a^0 =e, a^1 =a,
an=a(an−^1 ), a−n=a−^1 (a−^1 )n−^1 ifn> 1.It is readily verified that, for allm,n∈Z,
aman=am+n,(am)n=amn.The set〈a〉={an:n∈Z}is a commutative subgroup ofG,thecyclic subgroup gen-
erated by a. Evidently〈a〉containsaand is contained in every subgroup ofGwhich
containsa.
If we regardZas a group under addition, then the mappingn→anis a homomor-
phism ofZonto〈a〉. Consequently〈a〉is isomorphic to the factor groupZ/N,where
Nis the subgroup ofZconsisting of all integersnsuch thatan=e. Evidently 0∈N,
andn∈Nimplies−n∈N. Thus eitherN={ 0 }orNcontains a positive integer.
In the latter case, letsbe the least positive integer inN. By Proposition 14, for any
integernthere exist integersq,rsuch that
n=qs+r, 0 ≤r<s.Ifn∈N,thenalsor=n−qs∈Nand hencer=0, by the definition ofs. It follows
thatN=sZis the subgroup ofZconsisting of all multiples ofs. Thus either〈a〉
is isomorphic toZ, and is an infinite group, or〈a〉is isomorphic to the factor group
Z/sZ, and is a finite group of orders. We say that the elementaitself is ofinfinite
orderif〈a〉is infinite and oforder sif〈a〉is of orders.
It is easily seen that in acommutativegroup the set of all elements of finite order
is a subgroup, called itstorsion subgroup.
IfSis any nonempty subset of a groupG, then the set〈S〉of all finite products
aε 11 a 2 ε^1 ···anεn,wheren∈N,aj ∈Sandεj =±1, is a subgroup ofG, called the