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202
3. PRODUCTS OF GROUPS
Theorem 3.6.14.
(Primary decomposition theorem) Let
G
be a ﬁnite
abelian group and let
p
1
;:::;p
s
be the primes dividing
j
G
j
.
Then
G
Š
GŒp
1
Ł
± ²²² ±
GŒp
s
Ł
.
Proof.
Let
n
D
p
k
1
1
p
k
2
s
²²²
p
k
s
s
be the prime decomposition of
n
D j
G
j
.
Applying the previous proposition with
˛
i
D
p
k
i
i
gives the result.
n
The decomposition of Theorem
3.6.14
is called the
primary decompo
sition
of
G
.
Corollary 3.6.15.
(Sylow’s theorem for Finite Abelian Groups) If
G
is a
ﬁnite abelian group, then for each prime
p
, the order of
GŒpŁ
is the largest
power of
p
dividing
j
G
j
. Moreover, any subgroup of
G
whose order is a
power of
p
is contained in
GŒpŁ
.
Proof.
By Corollary
3.6.12
, the order of
GŒpŁ
is a power of
p
. Since
j
G
j D
Q
p
j
GŒpŁ
j
, it follows that
j
GŒpŁ
j
is the largest power of
p
dividing
j
G
j
. If
A
is a subgroup of
G
whose order is a power of
p
, then
A
is a
p
–group, so
A
³
GŒpŁ
, by deﬁnition of
GŒpŁ
.
n
The primary decomposition and the Chinese remainder theorem.
For
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 Fall '08
 EVERAGE
 Algebra

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