8.4 Group homomorphisms and isomorphisms 199

to show that ker(ρ) is trivial; that is, it su¬ces to show that for all h1 ∈ H1

and h2 ∈ H2 , h1 + h2 = 0G implies h1 = 0G and h2 = 0G . But h1 + h2 = 0G

implies h1 = ’h2 ∈ H2 , and hence h1 ∈ H1 © H2 = {0G }, and so h1 = 0G .

Similarly, one shows that h2 = 0G , and that ¬nishes the proof. 2

Example 8.41. For n ≥ 1, the natural map ρ from Z to Zn sends a ∈ Z to

the residue class [a]n . This map is a surjective group homomorphism with

kernel nZ. 2

Example 8.42. We may restate the Chinese remainder theorem (Theo-

rem 2.8) in more algebraic terms. Let n1 , . . . , nk be pairwise relatively

prime, positive integers. Consider the map from the group Z to the group

Zn1 — · · · — Znk that sends x ∈ Z to ([x]n1 , . . . , [x]nk ). It is easy to see that

this map is a group homomorphism (this follows from Example 8.41 and

Theorem 8.22). In our new language, the Chinese remainder theorem says

that this group homomorphism is surjective and that the kernel is nZ, where

n = k ni . Therefore, by Theorem 8.26, the map that sends [x]n ∈ Zn

i=1

to ([x]n1 , . . . , [x]nk ) is a group isomorphism of the group Zn with the group

Zn1 — · · · — Znk . 2

Example 8.43. Let n1 , n2 be positive integers with n1 > 1 and n1 | n2 .

Then the map ρ : Zn2 ’ Zn1 that sends [a]n2 to [a]n1 is a surjective group

¯

homomorphism, and [a]n2 ∈ ker(¯) if and only if n1 | a; that is, ker(¯) =

ρ ρ

n1 Zn2 . The map ρ can also be viewed as the map obtained by applying

¯

Theorem 8.27 with the natural map ρ from Z to Zn1 and the subgroup n2 Z

of Z, which is contained in ker(ρ) = n1 Z. 2

Example 8.44. Let us reconsider Example 8.21. Let n be a positive in-

teger, let m ∈ Z, and consider the subgroup mZn of the additive group

Zn . Let ρ1 : Z ’ Zn be the natural map, and let ρ2 : Zn ’ Zn be the

m-multiplication map. The composed map ρ = ρ2 —¦ ρ1 from Z to Zn is also

a group homomorphism. The kernel of ρ consists of those integers a such

that am ≡ 0 (mod n), and so Theorem 2.7 implies that ker(ρ) = (n/d)Z,

where d := gcd(m, n). The image of ρ is mZn . Theorem 8.26 therefore

implies that the map ρ : Zn/d ’ mZn that sends [a]n/d to [ma]n is a group

¯

isomorphism. 2

Exercise 8.10. Verify that the “is isomorphic to” relation on abelian groups

is an equivalence relation; that is, for all abelian groups G1 , G2 , G3 , we have:

(a) G1 ∼ G1 ;

=

(b) G1 ∼ G2 implies G2 ∼ G1 ;

= =

200 Abelian groups

(c) G1 ∼ G2 and G2 ∼ G3 implies G1 ∼ G3 .

= = =

Exercise 8.11. Let G1 , G2 be abelian groups, and let ρ : G1 — G2 ’ G1

be the map that sends (a1 , a2 ) ∈ G1 — G2 to a1 ∈ G1 . Show that ρ is a

surjective group homomorphism whose kernel is {0G1 } — G2 .

Exercise 8.12. Suppose that G, G1 , and G2 are abelian groups, and that

ρ : G1 — G2 ’ G is a group isomorphism. Let H1 := ρ(G1 — {0G2 }) and

H2 := ρ({0G1 } — G2 ). Show that

(a) H1 and H2 are subgroups of G,

(b) H1 + H2 = G, and

(c) H1 © H2 = {0G }.

Exercise 8.13. Let ρ be a group homomorphism from G into G . Show

that for any subgroup H of G, we have ρ’1 (ρ(H)) = H + ker(ρ).

Exercise 8.14. Let ρ be a group homomorphism from G into G . Show

that the subgroups of G containing ker(ρ) are in one-to-one correspondence

with the subgroups of img(ρ), where the subgroup H of G containing ker(ρ)

corresponds to the subgroup ρ(H) of img(ρ).

Exercise 8.15. Let G be an abelian group with subgroups H ⊆ H .

(a) Show that we have a group isomorphism

G/H

G/H ∼ .

=

H /H

(b) Show that if [G : H] is ¬nite (even though G itself may have in¬nite

order), then [G : H] = [G : H ] · [H : H].

Exercise 8.16. Show that if G = G1 — G2 for abelian groups G1 and G2 ,

and H1 is a subgroup of G1 and H2 is a subgroup of G2 , then G/(H1 —H2 ) ∼

=

G1 /H1 — G2 /H2 .

Exercise 8.17. Let ρ1 and ρ2 be group homomorphisms from G into G .

Show that the map ρ : G ’ G that sends a ∈ G to ρ1 (a) + ρ2 (a) ∈ G is

also a group homomorphism.

Exercise 8.18. Let G and G be abelian groups. Consider the set H of all

group homomorphisms ρ : G ’ G . This set is non-empty, since the map

that sends everything in G to 0G is trivially an element of H. We may de¬ne

an addition operation on H as follows: for ρ1 , ρ2 ∈ H, let ρ1 + ρ2 be the map

ρ : G ’ G that sends a ∈ G to ρ1 (a) + ρ2 (a). By the previous exercise, ρ is

8.4 Group homomorphisms and isomorphisms 201

also in H, and so this addition operation is a well-de¬ned binary operation

on H. Show that H, together with this addition operation, forms an abelian

group.

Exercise 8.19. This exercise develops an alternative, “quick and dirty”

proof of the Chinese remainder theorem, based on group theory and a count-

ing argument. Let n1 , . . . , nk be pairwise relatively prime, positive integers,

and let n := n1 · · · nk . Consider the map ρ : Z ’ Zn1 — · · · — Znk that sends

x ∈ Z to ([x]n1 , . . . , [x]nk ).

(a) Using the results of Example 8.41 and Theorem 8.22, show (directly)

that ρ is a group homomorphism with kernel nZ.

(b) Using Theorem 8.26, conclude that the map ρ given by that theorem,

¯

which sends [x]n to ([x]n1 , . . . , [x]nk ), is an injective group homomor-

phism from Zn into Zn1 — · · · — Znk .

(c) Since |Zn | = n = |Zn1 — · · · — Znk |, conclude that the map ρ is¯

surjective, and so is an isomorphism between Zn and Zn1 — · · · — Znk .

Although simple, this proof does not give us an explicit formula for comput-

ing ρ’1 .

¯

Exercise 8.20. Let p be an odd prime; consider the squaring map on Z— .

p

(a) Using Exercise 2.5, show that the kernel of the squaring map on Z—

p

consists of the two elements [±1]p .

(b) Using the results of this section, conclude that there are (p ’ 1)/2

squares in Z— , each of which has precisely two square roots in Z— .

p p

Exercise 8.21. Consider the group homomorphism ρ : Z — Z — Z ’ Q—

that sends (a, b, c) to 2a 3b 12c . Describe the image and kernel of ρ.

Exercise 8.22. This exercise develops some simple ” but extremely use-

ful” connections between group theory and probability theory. Let ρ : G ’

G be a group homomorphism, where G and G are ¬nite abelian groups.

(a) Show that if g is a random variable with the uniform distribution on

G, then ρ(g) is a random variable with the uniform distribution on

img(ρ).

(b) Show that if g is a random variable with the uniform distribution

on G, and g is a ¬xed element in img(ρ), then the conditional dis-

tribution of g, given that ρ(g) = g , is the uniform distribution on

ρ’1 ({g }).

(c) Show that if g1 is a ¬xed element of G , g1 is uniformly distributed

202 Abelian groups

over ρ’1 ({g1 }), g2 is a ¬xed element of G , and g2 is a ¬xed element of

ρ’1 ({g2 }), then g1 + g2 is uniformly distributed over ρ’1 ({g1 + g2 }).

(d) Show that if g1 is a ¬xed element of G , g1 is uniformly distributed

over ρ’1 ({g1 }), g2 is a ¬xed element of G , g2 is uniformly distributed

over ρ’1 ({g2 }), and g1 and g2 are independent, then g1 + g2 is uni-

formly distributed over ρ’1 ({g1 + g2 }).