3.6. FINITELY GENERATED ABELIAN GROUPS
197
A
D
G
tor
. Consequently,
B
Š
G=G
tor
, so the rank of
B
is determined.
This proves part (b).
For part (c), note that any free module is torsion free. On the other
hand, if
G
is torsion free, then by the decomposition of part (b),
G
is
free.
n
Lemma 3.6.4.
Let
x
be a torsion element in an abelian group, with order
a
and let
p
be a prime number.
(a)
If
p
divides
a
, then
Z
x=p
Z
x
Š
Z
p
.
(b)
If
p
does not divide
a
, then
p
Z
x
D
Z
x
.
Proof.
Consider the group homomorphism of
Z
onto
Z
x
,
r
7!
r x
, which
has kernel
a
Z
. If
p
divides
a
, then
pZ
aZ
, and the image of
p
Z
in
Z
x
is
p
Z
x
. Hence by Proposition
2.7.13
,
Z
=.p/
Š
Z
x=p
Z
x
. If
p
does
not divide
a
, then
p
and
a
are relatively prime. Hence there exist integers
s; t
such that
sp
C
ta
D
1
. Therefore, for all integers
r
,
r x
D
1rx
D
psrx
C
tarx
D
psrx
(since
ax
D
0
). It follows that
Z
x
D
p
Z
x
.
n
Lemma 3.6.5.
Suppose
G
is an abelian group,
p
is a prime number. and
pG
D f
0
g
. Then
G
is a vector space over
Z
p
. Moreover, if
'
W
G
!
G
is a surjective group homomorphism, then
G
is an
Z
p
–vector space as
well, and
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Abelian group, Zp –vector space, Zx Š Zp

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