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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 ΠG. 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 ΠG.
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 ΠG. 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
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