196
3. PRODUCTS OF GROUPS
(a)
G
is a direct product of cyclic groups,
G
Š
Z
a
1
±
Z
a
2
± ²²² ±
Z
a
s
±
Z
k
;
where
a
i
³
2
, and
a
i
divides
a
j
for
i
´
j
.
(b)
The decomposition in part (a) is unique, in the following sense:
If
G
Š
Z
b
1
±
Z
b
2
± ²²² ±
Z
b
t
±
Z
`
;
where
b
j
³
2
, and
b
i
divides
b
j
for
i
´
j
, then
`
D
k
,
s
D
t
,
and
a
j
D
b
j
for all
j
.
An element of ﬁnite order in an abelian group
G
is also called a
torsion
element
. It is easy to see that an integer linear combination of torsion ele
ments is a torsion element, so the set of torsion elements forms a subgroup,
the torsion subgroup
G
tor
. We say that
G
is a
torsion group
if
G
D
G
tor
and that
G
is
torsion free
if
G
tor
D f
0
g
. It is easy to see that
G=G
tor
is
torsion free. See Exercise
3.6.1
.
An abelian group is ﬁnite if, and only if, it is a ﬁnitely generated tor
sion group. (See Exercise
3.6.4
.) Note that if
G
is ﬁnite, then ann
.G/
D
f
r
2
Z
W
rx
D
0
for all
x
2
G
g
is a nonzero subgroup of
Z
(Exercise
3.6.5
). Deﬁne the
period
of
G
to be the least positive element of ann
.G/
.
If
a
is the period of
G
, then ann
.G/
D
a
Z
, since a subgroup of
Z
is always
generated by its least positive element.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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