Quotient Groups and Homomorphisms

Recall that for N, a normal subgroup of a group G, whenever

†

a ≡ b(mod N ) and

†

c ≡ d(mod N ), then

†

ac ≡ bd(mod N ). Recall also that

†

a ≡ b(mod N ) if and only if

†

Na = Nb . Putting these two results together, we see that if

†

Na = Nb and

†

Nc = Nd , then

†

Nac = Nbd . This means, of course, we can define a product on the set of right cosets ofN in G by

†

NaNc = Nac (or equivalently on the set of congruence classes of G modulo Nby

†

a[ ] c[ ] = ac[ ] ) and such a product is well-defined.

Theorem: Let G/N denote the set of all right cosets of N in G under the operation

†

NaNb = Nab . G/N is a group, called the quotient group (or factor group) of G by N.

Recall that [G:N], the index of N in G, is the number of right cosets of N in G and

consequently |G/N| =[G:N]. It follows that when G is finite, |G/N| =

†

| G|| N |

.

Problem 1: Prove that if G is abelian and N is a subgroup of G, then N is a normalsubgroup and G/N is an abelian group.

The converse is false, as the next problem demonstrates.

Problem 2: In

†

D4 , let r be the transformation defined by rotating

†

p2

units about the z-

axis. We showed in the last handout that

†

K ={e,r,r 2,r3} is a normal subgroup. Showthat

†

D4 /K is abelian.

However, in some cases, knowing information about N and G/N can give us someinformation about G.

Theorem: If G is a group with center Z(G) such that the quotient group G/Z(G) is cyclic,then G is abelian.

Problem 3: Prove that if N is a normal subgroup of a group G and if every element of Nand G/N has finite order, then every element of G has finite order.

So what is the relation between quotient groups and homomorphisms? Before answering,we will first recall the definition of a homomorphism.

Definition: Let G and H be groups with operations denoted by

†

* and

†

•, respectively. Ahomomorphism is a map

†

f :G Æ H satisfying

†

f (a* b)= f (a)• f (b) for all

†

a,b ΠG .

Examples:(1) Let Z be the group of integers under addition and

†

Zn be the group of integers modulon. Define

†

g :Z Æ Zn by sending each integer a to the class of a modulo n. Show that gis a homomorphism. Is g injective? Is g surjective?

(2) Let C denote the set of non-zero complex numbers

†

a + bi under multiplication.

Define

†

f :C Æ GL(2,R) by

†

f (a + bi)=a -bb a

È

Î Í ˘

˚ ˙ . Show that f is a homomorphism. Is f

injective? Is f surjective?

Definition: A homomorphism that is also a bijection (one-to-one and onto) is called anisomorphism. If there is an isomorphism

†

f :G Æ H , we say that G is isomorphic to Hand write

†

G @ H .

Examples:(1) Let

†

G =< a >be a cyclic group of order n and define

†

f :G Æ Zn by

†

f (ai )= i[ ] . It isclear that f is a bijection and furthermore

†

f (aia j )= f (ai + j )= i + j[ ] = i[ ] + j[ ] = f (ai)+ f (a j ), so f is an isomorphism. Thus everycyclic group of order n is isomorphic to

†

Zn .

(2) Let R be the group of real numbers under addition and

†

R+ be the group of positivereal numbers under multiplication. Define

†

f :R Æ R+ by

†

f (a)= 2a . Show f is anisomorphism.

Problem 4: (a) Let

†

R+ denote the group of positive real numbers under multiplication andR be the group of all real numbers under addition. Define

†

f :R+ Æ R by

†

f (a)= ln(a).Show that f is an isomorphism.

(b) Show that

†

U9 is isomorphic

†

Z6.

We are now ready to explore the relationship between quotient groups andhomomorphisms. The following lemma shows that each normal subgroup N of a groupG provides us with a homomorphism.

Lemma: Let G be a group and N a normal subgroup of G. Define

†

p :G Æ G/N by

†

p(g)= Ng . Then

†

p is a homomorphism from G onto G/N .

Conversely, each homomorphism provides us with a quotient group. To see that,however, we need another definition.

Definition: Let

†

j :G Æ H be a homomorphism of groups. The kernel of

†

j is the set

†

Kj ={g ΠG| j(g)= eH}. In other words, the kernel of a homomorphism is the set ofelements in G that are mapped to the identity element in H.

Example: Let Z be the group of integers under addition and

†

Zn be the group of integersmodulo n. Define

†

g :Z Æ Zn by sending each integer a to the class of a modulo n. Weshowed in a previous example that g is a homomorphism. Find the kernel of g.

We claim that

†

Kj is a normal subgroup of G. In order to establish this fact, the followinglemma will be useful.

Lemma: Let

†

j :G Æ H be a homomorphism of groups. Then(i)

†

j(e)= eH

(ii)

†

j(g-1)= j(g)( )-1 for all

†

g ΠG .

Theorem: Let

†

j :G Æ H be a homomorphism of groups with kernel

†

Kj . Then

†

Kj is anormal subgroup of G.

The previous theorem makes it clear that each homomorphism provides us with aquotient group, namely

†

G/Kj . We will explore this quotient group further.

Note that if

†

j :G Æ H is injective, then by the first part of the lemma above, it is clearthat

†

Kj ={e}. What is less obvious is that the converse is true as well.

Lemma: A homomorphism

†

j :G Æ H is injective if and only if

†

Kj ={e}.

This lemma makes it especially easy to check whether homomorphisms are injective.

Example: Let G be a group and let

†

a ΠG be a fixed element. Define

†

j :G Æ G by

†

j(g)= aga-1. Show that

†

j is an isomorphism. (Note: An isomorphism from a group Gto itself is called an automorphism. This particular automorphism is called the innerautormorphism of G induced by a.)

Problem 5: Let

†

f :G Æ H and

†

g :H Æ K be isomorphisms of groups. Prove that thecomposite function

†

g o f :G Æ K is also an isomorphism.

Let’s consider the question of which groups can arise as the homomorphic images of agiven group G. In other words, given a group G, can we find all groups H such that thereis a homomorphism from G onto H? We showed above that for each normal subgroup Nof G, the quotient group G/N is a homomorphic image of G, since we have the surjectivehomomorphism

†

p :G Æ G/N defined by

†

p(g)= Ng . The theorem below shows that upto isomorphism, these are the only groups which are homomorphic images of G.

Theorem: Let

†

j :G Æ H be a surjective homomorphism with kernel

†

Kj . Then thequotient group

†

G/Kj is isomorphic to H. More particularly, there is an isomorphism

†

y :G/Kj Æ H such that

†

y o p = j (where

†

p is the map defined in the lemma followingproblem 4.)

Example: Consider the map

†

j :Z30 Æ Z5 defined by sending the equivalence class ofeach integer k modulo 30 to the class of k modulo 5:

†

j k[ ]30( ) = k[ ]5. Show

†

j is asurjective homomorphism, describe its kernel

†

Kj , and construct the isomorphism

†

y :Z30 /Kj Æ Z5.

Note that as a consequence of this theorem, we see that for each homomorphic image ofG, there is a normal subgroup, and conversely for each normal subgroup, there is ahomomorphic image of G. Thus there is a one-to-one correspondence betweenhomomorphic images of G and normal subgroups of G.

Problem 6: Find all homomorphic images (up to isomorphism) of

†

Z12 .