186
3. PRODUCTS OF GROUPS
'
W
.a
1
;:::;a
n
/
7!
P
i
a
i
x
i
. The kernel
N
of
'
is a subgroup of
Z
n
, and
G
Š
Z
n
=N
. Conversely, for any subgroup
N
of
Z
n
,
Z
n
=N
is a ﬁnitely
generated abelian group.
Therefore, to understand ﬁnitely generated abelian groups, we must
understand subgroups of
Z
n
. The study of subgroups of
Z
n
involves linear
algebra over
Z
.
The theory presented in this section and the next is a special case of
the structure theory for ﬁnitely generated modules over a principal ideal
domain, which is discussed in sections
8.4
and
8.5
, beginning on page
369
. The reader or instructor who needs to save time may therefore prefer
to omit some of the proofs in Sections
3.5
and
3.6
in the expectation of
treating the general case in detail.
We deﬁne linear independence in abelian groups as for vector spaces: a
subset
S
of an abelian group
G
is linearly independent over
Z
if whenever
x
1
;:::;x
n
are
distinct
elements of
S
and
r
1
;:::;r
n
are elements of
Z
, if
r
1
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, Ring, Abelian group

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