Graphics Reference
In-Depth Information
(
)
+¢
(
)
=+¢
(
)
gg
,
,...,
g
gg
,
¢
,...,
g
¢
g gg
,
+¢
g
,...,
g
+¢
g
.
12
n
12
n
1
12
2
n
n
It is easy to check that (G
1
¥ G
2
¥ ···¥ G
n
,+) is an abelian group. Except for the
different name and notation that is used in the abelian case, the external direct sum
is just the direct product defined at the end of the last section.
The groups
Z
n
and
R
n
B.5.6. Example.
are the external direct sums of n copies of
Z
and
R
, respectively.
Definition.
Let G
1
, G
2
,..., and G
n
be subgroups of an abelian group G. Define the
sum
of the groups G
i
, denoted by G
1
+ G
2
+ ···+ G
n
, by
{
}
.
GG
+
+◊◊◊+
G
=
gg
+
+◊◊◊+
ggG
Œ
1
2
n
1
2
n
i
i
It is easy to show that G
1
+ G
2
+ ···+ G
n
is a subgroup of G.
Definition.
If G
1
, G
2
,..., and G
n
are subgroups of a given abelian group G and if
each element g in G can be written uniquely in the form g
1
+g
2
+ ···+g
n
with
g
i
in G
i
, then G is called the
internal direct sum
of the groups G
i
and we write
G = G
1
≈G
2
≈ ···≈G
n
.
Because it is easily shown that G
1
≈G
2
≈ ···≈G
n
is isomorphic to
G
1
ƒG
2
ƒ ···ƒG
n
, one usually does not distinguish between internal and external
direct sums and uses the same symbol ≈ to denote either direct sum. The context
decides which is being used. One also uses the summation notation, so that the
expressions
n
G
≈
G
≈◊◊◊≈
G
,
G
ƒ
G
ƒ◊◊◊ƒ
G
,
and
=
G
1
2
n
1
2
n
i
i
1
will all refer to the same object.
B.5.7. Theorem.
(The Fundamental Theorem of Finitely Generated Abelian
Groups) Let G be a finitely generated abelian group. Then
G
ª
ZZ
≈
≈◊◊◊≈
ZZ
≈
≈
Z
≈◊◊◊≈
Z
,
n
n
n
t
12
44
44
3
1
2
r
for some r, t ≥ 0, where 1 < n
i
Œ
Z
and n
i
| n
i+1
if t > 0. The integer r is called the
rank
of
G and is denoted by rank(G). The integers n
1
, n
2
,..., and n
t
are called the
torsion coef-
ficients
of G. Both the rank and the torsion coefficients are uniquely determined by G.
Proof.
See [Dean66].
Note that the example
ZZZ
Ż
2
3
6
