✲ ◗
◗ ◗
◗ ◗
◗ ◗
◗ ◗
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,
ZZ
6
5 2
4 2
4 2
4 1
3 5
2 4
4 2
1 3
5 3
1 3
5 1
5 1
3 5
3 1
3 5
1 5
1 3
4 2
4 2
4 2
Fig. 3.6. Z
6
{0, 2, 4} is isomorphic to Z
2
.
we observe that it is isomorphic to Z
3
, the only group of order 3. Therefore, Z
6
{0, 3} ∼ = Z
3
. Moreover, {0, 3} with binary operation +
6
is isomorphic to Z
2
, as one may see from the top left corner of the table for Z
6
. So, we have Z
6
Z
2
∼ = Z
3
. Similarly, Z
6
Z
3
∼ = Z
2
, as shown in Figure 3.6. For a beginner, factor groups seem to be one of the hardest concepts in
group theory. Given a factor group G H, the key idea to remember is
that each element of the factor group has the form aH: It is a set, a coset of H. Now, we could represent each element of a factor group with a representative
from the coset. For example, the element 4 could represent the coset {1, 4} for
factor group Z
6
{0, 3}. However, don’t forget that this element is congruent to 1 modulo
{0, 3}.
3.3 Advanced Structures
In this section, we delve into advanced algebra by looking at increasingly rich algebraic structures we will encounter in our study of homology. Our goal in
this section is to generalize Theorem 3.10, first to modules and then to graded modules.
3.3.1 Free Abelian Groups
Recall that a finitely generated Abelian group is isomorphic to a product of infinite and finite cyclic groups. In this section, we will characterize infinite
factors using the notion of free Abelian groups. As we will only deal with Abelian groups, we will use
+ to denote the group operation and 0 for the identity element. For n
∈ Z
+
, a ∈ G, we use na = a + a + · · · + a and −na = −a + −a + · · · + −a to denote the sum of n copies of a and its inverse,
respectively. Finally, 0a = 0, where the first 0 is in Z, and the second is in G.
It is important to realize that G is still a group with a single group operation, addition, even though we use multiplication in our notation. We shall shift
our view later in defining modules and vector spaces. Let us start with two equivalent conditions.
Theorem 3.13 Let X be a subset of a nonzero Abelian group G. The following conditions on X are equivalent.
a Each nonzero element a in G can be uniquely expressed in the form a
= n
1
x
1
+ n
2
x
2
+ · · · + n
r
x
r
for n
i
= 0 in Z and distinct x
i
∈ X. b X generates G, and n
1
x
1
+ n
2
x
2
+ · · · + n
r
x
r
= 0 for n
i
∈ Z and x
i
∈ X iff n
1
= n
2
= · · · = n
r
= 0. The conditions should remind the reader of linearly independent vectors. As
we will soon find out, this similarity is not accidental.
Definition 3.19 free Abelian group An Abelian group having a nonempty generating set X satisfying the conditions in Theorem 3.13 is a free Abelian
group and X is a basis for the group.
We have already seen a free Abelian group: The finite direct product of the group Z with itself is a free Abelian group with a natural basis. In fact, we
may use this group as a prototype.
Theorem 3.14 If G is a nonzero free Abelian group with a basis of r elements, then G is isomorphic to Z
× Z × · · · × Z for r factors. Furthermore, while we may form different bases for a free Abelian group, all
of them will have the same size.
Theorem 3.15 rank Let G be a nonzero free Abelian group with a finite ba- sis. Then, every basis of G is finite and all bases have the same number of
elements, the rank of G, rank G
= log
2
|G2G|. Subgroups of free Abelian groups are simply smaller free Abelian groups.
Theorem 3.16 A subgroup K of a free Abelian group G with finite rank n is a free Abelian group of rank s
≤ n. Furthermore, there exists a basis {x
1
, x
2
, . . . , x
n
} for G and d
1
, d
2
, . . . , d
s
∈ Z
+
, such that {d
1
x
1
, d
2
x
2
, . . . , d
s
x
s
} is a basis for K.
