Graphics Reference
In-Depth Information
B.5
Abelian Groups
In this section we concentrate on abelian groups and we shall use the standard addi-
tive notation.
Definition.
Let G be an abelian group and let g
1
, g
2
,..., g
k
Œ G. Then
{
}
ZZ
gg
+
+◊◊◊+
Z
gngng
=
+
+◊◊◊+
ngn
Œ
Z
1
2
k
1
1
2
2
k
k
i
is called the
subgroup
of G
generated by
the g
1
, g
2
,..., and g
k
.
It is easy to show that
Z
g
1
+
Z
g
2
+ ···+
Z
g
k
is in fact a subgroup of G and also that it
is the intersection of all subgroups of G which contain the elements g
1
, g
2
,..., and g
k
.
B.5.1. Lemma.
For any abelian group G the elements of finite order form a unique
subgroup T(G) called the
torsion subgroup
of G.
Proof.
The proof is clear since o(0) = 1, o(-g) = o(g), and o(g + h) | o(g)o(h) for all
g, h Œ G.
Definition.
An abelian group G is said to be
torsion-free
if it has no element of finite
order other than 0, that is, T(G) =
0.
Clearly, G/T(G) is a torsion-free group for every abelian group G.
Definition.
An abelian group G is said to be
finitely generated
if
GZg Zg
=+
◊ ◊ ◊ +
Zg
n
1
2
for some g
1
, g
2
,..., g
n
Œ G. In that case, the g
i
are called
generators
for G.
B.5.2. Example.
All cyclic groups are finitely generated.
The group
Z
2
B.5.3. Example.
is not cyclic, but it is finitely generated since
Z
2
=
Z
(1,0) +
Z
(0,1). A similar statement holds for
Z
n
, n ≥ 2 .
B.5.4. Example.
The groups
Q
and
R
are not finitely generated.
B.5.5. Lemma.
Subgroups of finitely generated abelian groups are finitely
generated.
Proof.
See [Dean66].
Definition.
Let G
1
, G
2
,..., and G
n
be abelian groups. The
external direct sum
of the
G
i
, denoted by G
1
ƒ G
2
ƒ ···ƒ G
n
, is the group (G
1
¥ G
2
¥ ···¥ G
n
,+), where the oper-
ation + is defined by