188
3. PRODUCTS OF GROUPS
combination of finitely many elements of
B
. Since
S
is finite, it is con
tained in the subgroup geneated by a finite subset
B
0
of
B
.
But then
G
D
Z
S
Z
B
0
.
So
B
0
generates
G
.
It follows from the previous
lemma that
B
0
D
B
.
n
Proposition 3.5.5.
Any two bases of a finitely generated free abelian group
have the same cardinality.
Proof.
Let
G
be a finitely generated free abelian group. By the previous
lemma, any basis of
G
is finite. If
G
has a basis with
n
elements, then
G
Š
Z
n
, by Proposition
3.5.2
. Since
Z
n
and
Z
m
are nonisomorphic if
m
¤
n
,
G
cannot have bases of different cardinalities.
n
Definition 3.5.6.
The
rank
of a finitely generated free abelian group is the
cardinality of any basis.
Proposition 3.5.7.
Every subgroup of
Z
n
can be generated by no more
than
n
elements.
Proof.
The proof goes by induction on
n
. We know that every subgroup of
Z
is cyclic (Proposition
2.2.21
), so this takes care of the base case
n
D
1
.
Suppose that
n > 1
and that the assertion holds for subgroups of
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 Fall '08
 EVERAGE
 Algebra, Abelian group, free abelian group, Zm

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