Investiga tion 30

Homomorphisms and Isomorphism

Theorems

Focus Questions

By the end of this investigation, you should be able to give precise and thorough

answers to the questions listed below. You may want to keep these questions in mind

to focus your tho ughts as you complete the investigation.

• What is a homomorphism of groups, and how is a homomorphism differe nt than

an isomorphism?

• What ar e monomorphisms an d epimorphisms of groups?

• What are the kernel and image of a gr oup homomorphism, and what properties do

they satisfy?

• What are the isomorphism theorems for groups, and h ow do they use homo mor-

phisms to establish rela tionships between groups?

Preview Activity 30 .1. As we saw in Investigation 29, the notion of isomorphism formalizes what

it means for two groups to be essentially the same. Recall that an isomorphism of groups is a

bijective, structure-preserving fun c tion. In group theory, structure-preserving maps are important

even if they are not bijections. In this activity, we will explo re three different kinds of structure-

preserving f unctions. (Throughout the activity, recall that we use the notation [k]

n

to d enote the

congruence class of k in Z

n

.)

(a) Is the fun ction ϕ : Z

3

→ Z

6

deﬁned by ϕ([k]

3

) = [4k]

6

structure-preserving? Is ϕ an

injection? Is ϕ a surjection? Verif y your answers. (You may assume that ϕ is well-d e ﬁned.)

(b) Is the function ϕ : Z

6

→ Z

3

deﬁned by ϕ([k]

6

) = [k]

3

structure-preserving? Is ϕ an

injection? Is ϕ a surjection? Verif y your answers. (You may assume that ϕ is well-d e ﬁned.)

(c) Is the fun ction ϕ : Z

6

→ Z

4

deﬁned by ϕ([k]

6

) = [2k]

4

structure-preserving? Is ϕ an

injection? Is ϕ a surjection? Verif y your answers. (You may assume that ϕ is well-d e ﬁned.)

419

420 Investigation 30. Homo morphisms and Isomorphism Theorems

Homomorphisms

Preview Activity 30.1 illustrates that it is possible to have stru c ture-preserving maps tha t are injec -

tions but not surjections, surjections but n ot injections, or neither surjectio ns nor injections. When

we study groups, we are mostly interested in maps that preserve the gro up structure or operation.

Such maps—whether they are injective, surjective, neither, or both—are called homomorphisms,

deﬁned formally as follows:

Deﬁnition 30.2. Let G and H be gr oups. A function ϕ from G to H is a homomorphism of groups

if

ϕ(ab) = ϕ(a)ϕ(b)

for all a, b ∈ G.

Just like isomorphism, the word homomorphism comes from two Greek words: homos, which

means similar or like, and morphe, which means form or structure. Thus, when there is a homo-

morphism from one group to another, it means tha t there is some similarity of structur e betwee n th e

two groups. Just like a n isomorphism, a homomorphism is an operation-preserving or a structure-

preserving fu nction, but not necessarily a bijectio n.

Although it is not a requirement, some homomorphisms ar e also injections, surjections, or bi-

jections (as seen in Preview Activity 30.1). Homomorphisms that satisfy these additional properties

are given special names. In particular,

• a monomorphism is an injective h omomorphism;

• an epimorphism is a surjective homomorphism; and

• an isomorphism is a bijective homomorphism.

If ϕ : G → G

′

is an epimorphism, we call G

′

a homomorphic image of G.

Activity 30.3. Determine whether each of the following functions is a homomorphism from G to

H. If a function is a homomorphism, decide wh ether it is a monomorphism, an epimorphism, an

isomorphism, or none of these.

(a) G = Z, H = Z

5

, and ϕ(k) = [k]

5

(b) G = Z

3

, H = Z

18

, and ϕ([k]

3

) = [6k]

18

(c) G = Z, H = Z

2

⊕ Z

4

, and ϕ(k) = ([k]

2

, [k]

4

)

(d) G = R

+

, H = R

+

, and ϕ(k) =

√

k

(e) G = U

12

, H = Z

6

, and ϕ([k]

12

) = [k]

6

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