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)
4 Direct Products & Finitely Generated Abelian Groups
In this short chapter we see a straightforward way to create new groups from old using the Cartesian
product.
Example 4.1. Given Z
2
= {0, 1}, the Cartesian product
Z
2
×Z
2
=
(0, 0), (0, 1), (1, 0), ( 1, 1)
has four elements. This set inherits a group structure in a natural way by adding co-ordinates
(x, y) + (v, w) := (x + v, y + w)
where x + v and y + w are computed in (Z
2
, +
2
). This is a binary operation on Z
2
× Z
2
, with a
familiar-looking table: it has exactly the same structure as the Cayley table for the Klein four-group!
+ (0, 0) (0, 1) (1, 0) (1, 1)
(0, 0) (0, 0) (0, 1) (1, 0) (1, 1)
(0, 1) (0, 1) (0, 0) (1, 1) (1, 0)
(1, 0) (1, 0) (1, 1) (0, 0) (0, 1)
(1, 1) (1, 1) (1, 0) (0, 1) (0, 0)
↭
◦ e a b c
e e a b c
a a e c b
b b c e a
c c b a e
We conclude that Z
2
×Z
2
∼
=
V is indeed a group.
This construction works in general.
Theorem 4.2 (Direct product). The natural component-wise operation on the Cartesian product
n
∏
k=1
G
k
= G
1
×··· × G
n
, (x
1
, . . . , x
n
) ·(y
1
, . . . , y
n
) := (x
1
y
1
, . . . , x
n
y
n
)
defines a group structure: the direct product. This is abelian if each G
k
is abelian.
The proof is a simple exercise. Being a Cartesian product, a direct product has order equal to the
product of the orders of its components
n
∏
k=1
G
k
=
n
∏
k=1
|
G
k
|
Examples 4.3. 1. Consider the direct product of groups (Z
2
, +
2
) and (Z
3
, +
3
):
Z
2
×Z
3
=
(0, 0), (0, 1), (0, 2), ( 1, 0), (1, 1), (1, 2)
This is abelian and has order 6, so we might guess that it is isomorphic to (Z
6
, +
6
). To see this
we need a generator: choose (1, 1) and observe that
⟨
(1, 1)
⟩
=
(1, 1), (0, 2), (1, 0), ( 0, 1), (1, 2), (0, 0)
= Z
2
×Z
3
The map ϕ(x) = (x, x) is therefore an isomorphism ϕ : Z
6
∼
=
Z
2
×Z
3
.
28