Theorem 8.18. Let G be a ¬nite abelian group and H a subgroup. Then

[G : H] = |G|/|H|. Moreover, if H is another subgroup of G with H ⊆ H ,

then

[G : H] = [G : H ][H : G].

Proof. The fact that [G : H] = |G|/|H| follows directly from Theorem 8.15.

The fact that [G : H] = [G : H ][H : G] follows from a simple calculation:

|G| |G|/|H| [G : H]

.2

[G : H ] = = =

|H | |H |/|H| [H : H]

Example 8.29. For the additive group of integers Z and the subgroup nZ

for n > 0, the quotient group Z/nZ is precisely the same as the additive

group Zn that we have already de¬ned. For n = 0, Z/nZ is essentially just

a “renaming” of Z. 2

Example 8.30. Let G := Z6 and H = 3G be the subgroup of G consisting

of the two elements {[0], [3]}. The cosets of H in G are ± := H = {[0], [3]},

β := [1] + H = {[1], [4]}, and γ := [2] + H = {[2], [5]}. If we write out an

addition table for G, grouping together elements in cosets of H in G, then

we also get an addition table for the quotient group G/H:

+ [0] [3] [1] [4] [2] [5]

[0] [0] [3] [1] [4] [2] [5]

[3] [3] [0] [4] [1] [5] [2]

[1] [1] [4] [2] [5] [3] [0]

[4] [4] [1] [5] [2] [0] [3]

[2] [2] [5] [3] [0] [4] [1]

[5] [5] [2] [0] [3] [1] [4]

This table illustrates quite graphically the point of Theorem 8.17: for any

two cosets, if we take any element from the ¬rst and add it to any element

of the second, we always end up in the same coset.

We can also write down just the addition table for G/H:

+ ± β γ

± ± β γ

β β γ ±

γ γ ± β

8.3 Cosets and quotient groups 193

Note that by replacing ± with [0]3 , β with [1]3 , and γ with [2]3 , the

addition table for G/H becomes the addition table for Z3 . In this sense, we

can view G/H as essentially just a “renaming” of Z3 . 2

Example 8.31. Let us return to Example 8.24. The group Z— , as we 15

— )2 of Z— has order 2. Therefore, the

saw, is of order 8. The subgroup (Z15 15

quotient group Z— /(Z— )2 has order 4. Indeed, the cosets are ±00 = {[1], [4]},

15 15

±01 = {[’1], [’4]}, ±10 = {[2], [’7]}, and ±11 = {[7], [’2]}. In the quotient

group, ±00 is the identity; moreover, we have

2 2 2

±01 = ±10 = ±11 = ±00

and

±01 ±10 = ±11 , ±10 ±11 = ±01 , ±01 ±11 = ±10 .

This completely describes the behavior of the group operation of the quotient

group. Note that this group is essentially just a “renaming” of the group

Z2 — Z2 . 2

Example 8.32. As we saw in Example 8.25, (Z— )2 = {[±1]}. Therefore,

5

— /(Z— )2 has order 2. The cosets of (Z— )2 in Z— are

the quotient group Z5 5 5 5

— /(Z— )2 , ± is the identity,

±0 = {[±1]} and ±1 = {[±2]}. In the group Z5 0

5

and ±1 is its own inverse, and we see that this group is essentially just a

“renaming” of Z2 . 2

Exercise 8.7. Let H be a subgroup of an abelian group G, and let a and

a be elements of G, with a ≡ a (mod H).

(a) Show that ’a ≡ ’a (mod H).

(b) Show that na ≡ na (mod H) for all n ∈ Z.

Exercise 8.8. Let G be an abelian group, and let ∼ be an equivalence

relation on G. Further, suppose that for all a, a , b ∈ G, if a ∼ a , then

a + b ∼ a + b. Let H := {a ∈ G : a ∼ 0G }. Show that H is a subgroup of

G, and that for all a, b ∈ G, we have a ∼ b if and only if a ≡ b (mod H).

Exercise 8.9. Let H be a subgroup of an abelian group G.

(a) Show that if H is a subgroup of G containing H, then H /H is a

subgroup of G/H.

(b) Show that if K is a subgroup of G/H, then the set H := {a ∈ G :

a + H ∈ K} is a subgroup of G containing H.

194 Abelian groups

8.4 Group homomorphisms and isomorphisms

De¬nition 8.19. A group homomorphism is a function ρ from an

abelian group G to an abelian group G such that ρ(a + b) = ρ(a) + ρ(b)

for all a, b ∈ G.

Note that in the equality ρ(a + b) = ρ(a) + ρ(b) in the above de¬nition,

the addition on the left-hand side is taking place in the group G while the

addition on the right-hand side is taking place in the group G .

Two sets play a critical role in understanding a group homomorphism

ρ : G ’ G . The ¬rst set is the image of ρ, that is, the set ρ(G) = {ρ(a) :

a ∈ G}. The second set is the kernel of ρ, de¬ned as the set of all elements

of G that are mapped to 0G by ρ, that is, the set ρ’1 ({0G }) = {a ∈ G :

ρ(a) = 0G }. We introduce the following notation for these sets: img(ρ)

denotes the image of ρ, and ker(ρ) denotes the kernel of ρ.

Example 8.33. For any abelian group G and any integer m, the map that

sends a ∈ G to ma ∈ G is clearly a group homomorphism from G into

G, since for a, b ∈ G, we have m(a + b) = ma + mb. The image of this

homomorphism is mG and the kernel is G{m}. We call this map the m-

multiplication map on G. If G is written multiplicatively, we call this

the m-power map on G, and its image is Gm . 2

Example 8.34. Consider the m-multiplication map on Zn . As we saw in

Example 8.21, if d := gcd(n, m), the image mZn of this map is a subgroup

of Zn of order n/d, while its kernel Zn {m} is a subgroup of order d. 2

Example 8.35. Let G be an abelian group and let a be a ¬xed element of

G. Let ρ : Z ’ G be the map that sends z ∈ Z to za ∈ G. It is easy to see

that this is group homomorphism, since

ρ(z + z ) = (z + z )a = za + z a = ρ(z) + ρ(z ). 2

Example 8.36. As a special case of the previous example, let n be a positive

integer and let ± be an element of Z— . Let ρ : Z ’ Z— be the group

n n

z ∈ Z— . If the multiplicative order of

homomorphism that sends z ∈ Z to ± n

± is equal to k, then as discussed in §2.5, the image of ρ consists of the k

distinct group elements ±0 , ±1 , . . . , ±k’1 . The kernel of ρ consists of those

integers a such that ±a = [1]n . Again by the discussion in §2.5, the kernel

of ρ is equal to kZ. 2

Example 8.37. We may generalize Example 8.35 as follows. Let G be an

abelian group, and let a1 , . . . , ak be ¬xed elements of G. Let ρ : Z—k ’ G

8.4 Group homomorphisms and isomorphisms 195

be the map that sends (z1 , . . . , zk ) ∈ Z—k to z1 a1 + · · · + zk ak ∈ G. The

reader may easily verify that ρ is a group homomorphism. 2

Example 8.38. As a special case of the previous example, let p1 , . . . , pk

be distinct primes, and let ρ : Z—k ’ Q— be the group homomorphism that

sends (z1 , . . . , zk ) ∈ Z—k to pz1 · · · pzk ∈ Q— . The image of ρ is the set of all

1 k

non-zero fractions whose numerator and denominator are divisible only by

the primes p1 , . . . , pk . The kernel of ρ contains only the all-zero tuple 0—k .

2

The following theorem summarizes some of the most important properties

of group homomorphisms.

Theorem 8.20. Let ρ be a group homomorphism from G to G .

(i) ρ(0G ) = 0G .

(ii) ρ(’a) = ’ρ(a) for all a ∈ G.

(iii) ρ(na) = nρ(a) for all n ∈ Z and a ∈ G.

(iv) For any subgroup H of G, ρ(H) is a subgroup of G .