except for the possible arrangement of factors; that is, the number of factors of Z is unique and the prime powers
p
i r
i
are unique. Note how the product is composed of a number of infinite and finite cyclic
group factors. Intuitively, the infinite part captures those generators that are “free” to generate as many elements as they wish. The finite or “torsion” part
captures generators with finite order.
Definition 3.17 Betti numbers, torsion The number of factors of Z in The- orem 3.10 is the Betti number
βG of G. The subscripts of the finite cyclic factors are called the torsion coefficients of G.
3.2.4 Factor Groups
We saw in Theorem 3.4 how the left and right cosets defined by the kernel of a homomorphism were the same. The cosets are also the same for any normal
subgroup H by definition. We would like to treat the cosets defined by H as individual elements of another smaller group. To do so, we first derive a binary
operation from the group operation of G.
Theorem 3.11 Let H be a subgroup of a group G. Then, left coset multiplica- tion is well defined by the equation
aHbH = abH, iff the left and right cosets coincide.
The multiplication is well defined because it does not depend on the elements a
, b chosen from the cosets. Using left coset multiplication as a binary opera- tion, we get new groups.
Corollary 3.2 Let H be a subgroup of G whose left and right cosets coin- cide. Then, the cosets of H form a group G
H under the binary operation aHbH = abH.
Definition 3.18 factor group The group G
H in Corollary 3.2 is the factor group
or quotient group of G modulo H. The elements in the same coset of H
are said to be congruent modulo H. We have already seen a factor group defined by the kernel of a homomorphism
ϕ. The factor group, namely Gker ϕ, is naturally isomorphic to ϕG.
Theorem 3.12 fundamental homomorphism Let
ϕ : G → G
′
be a group homomorphism with kernel H. Then
ϕG is a group and the map µ : GH →
✲ ◗
◗ ◗
◗ ◗
◗ ◗
◗ ◗
s ✑
✑ ✑
✑ ✑
✑ ✑
✑ ✑
✑ ✸
G H
ϕ = µγ ϕG
G
µ gH = ϕg
γg = gH
Fig. 3.4. The fundamental homomorphism theorem. H = ker ϕ, and µ is the natural
isomorphism, corresponding to homomorphism γ.
3 3
3 3
3 3
1 4
2 5
2 5
1 4
1 4
4 1
2 5
2 5
2 5
4 1
4 1
2 5
ZZ
6
5 2
4 1
3 3
4 2
5 1
Fig. 3.5. Z
6
{0, 3} is isomorphic to Z
3
.
ϕG given by µgH = ϕg is an isomorphism. If γ : G → GH is the homo- morphism given by
γg = gH, then for each g ∈ G we have ϕg = µγg. µ is the
natural or canonical isomorphism, and γ is the corresponding homomor-
phism. The relationship between
ϕ, µ and γ is shown in a commutative diagram in Fig- ure 3.4. Homology characterizes topology using factor groups whose structure
is finitely Abelian. So, it is imperative to gain a full understanding of this method before moving on.
Example 3.6 factoring Z
6
The cyclic group Z
6
, on the left, has {0, 3} as a
subgroup. As Z
6
is Abelian, {0, 3} is normal, so we may factor Z
6
using this subgroup, getting the cosets
{0, 3}, {1, 4}, and {2, 5}. Figure 3.5 shows the table for Z
6
, ordered and shaded according to the cosets. The shading pattern gives rise to a smaller group, shown on the right, where each coset is collapsed
to a single element. Comparing this new group to the structures in Table 3.2,
