2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS31
2.4
2.4.1
Direct Products and Finitely Generated Abelian
Groups
Direct Products
Def 2.43. 4 The Cartesian product of sets S1 , S2 , · · · , Sn is the set of all
ordered n-tuples (a1 , · · · , an ) where ai ∈ Si for i = 1, 2 · · · , n.
n
Y
Si
=
S1 × S2 × · · · × S n
i=1
:= {(a1 , a2 , · · · , an ) | ai ∈ Si for i = 1, 2, · · · , n}.
Caution: The notation (a1 , a2 , · · · , an ) here has different meaning from
the cycle notation in permutation groups.
Thm 2.44.QIf G1 , G2 , · · · , Gn are groups with the corresponding multiplications, then ni=1 Gi is a group under the following multiplication:
(a1 , a2 , · · · , an )(b1 , b2 , · · · , bn ) := (a1 b1 , a2 b2 , · · · , an bn )
for (a1 , a2 , · · · , an ) and (b1 , b2 , · · · , bn ) in
Qn
i=1 Gi .
Proof.
1. Closed: ai ∈ Gi , bi ∈ Gi ⇒ ai bi ∈ Gi ⇒ (a1 b1 , · · · , an bn ) ∈
Qn
i=1 Gi .
2. Associativity: By the associativity in each Gi .
3. Identity: Let ei be the identity in Gi . Then e := (e1 , e2 , · · · , en ) is
the identity of G.
−1
4. Inverse: The inverse of (a1 , · · · , an ) is (a−1
1 , · · · , an ).
Ex 2.45. R2 , R3 are abelian groups.
(Every finite dimensional real vector space is an abelian group that is isomorphic to certain Rn .)
2
Ex 2.46. Z2 = {(a, b) | a, b ∈ Z} is
Qan subgroup
Qnof R .
(If Hi ≤ Gi for i = 1, · · · , n, then i=1 Hi ≤ i=1 Gi ).
4
1st HW: 1, 2, 14, 34, 46
32
CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS
Ex 2.47. If ei is the identity of Gi for i = 1, · · · , n. Then
Gj ' {e1 } × {e2 } × · · · × {ej−1 } × Gj × {ej+1 } × · · · × {en } ≤
n
Y
Gi .
i=1
So every Gj is isomorphic to a subgroup of
Qn
i=1 Gi
for j = 1, · · · , n.
Ex 2.48. The Klein 4-group V is isomorphic to Z2 × Z2 , which is not cyclic.
Ex 2.49. Z2 × Z3 ' Z6 is cyclic, and (1, 1) is a generator.
Thm 2.50. Let m and n be positive integers.
1. If gcd(m, n) = 1 (i.e. m and n are relative prime), then Zm × Zn is
cyclic and is isomorphic to Zmn , and (1, 1) is a generator of Zm × Zn .
2. If gcd(m, n) 6= 1, then Zm × Zn is not cyclic.
Proof.
5
1. Suppose gcd(m, n) = 1. The sum of k copies of (1, 1) is equal to
(k mod m, k mod n). The smallest positive integer k that makes
(k mod m, k mod n) = (0, 0) is k = mn because m and n are relative prime. So the cyclic subgroup of (1, 1) in Zm × Zn has the order
mn = |Zm × Zn |. So (1, 1) generates Zm × Zn . It implies that Zm × Zn
is a cyclic group isomorphic to Zmn .
2. Suppose gcd(m, n) = d 6= 1. Given (a, b) ∈ Zm × Zn , the sum of k
copies of (a, b) is (ka mod m, kb mod n). Notice that both m and n
n
m
divide mn
d = m · d = d · n. Then
(
mn
mn
a mod m,
b mod n) = (0, 0).
d
d
So the cyclic subgroup generated by (a, b) has an order no more than
mn
d < mn = |Zm × Zn |. Thus Zm × Zn can not be generated by any
one of its elements, which implies that Zm × Zn is not cyclic.
Q
Cor 2.51. The group ni=1 Zmi is cyclic and is isomorphic to Zm1 m2 ···mn if
and only if the numbers mi for i = 1, · · · , n are pairwise relative prime, that
is, the gcd of any two of them is 1.
5
1st HW: 1, 2, 14, 34, 46
2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS33
Proof. (For reference only) If mi for i = 1, · · · , n are pairwise relative prime,
then
!
!
n
n
n
Y
Y
Y
Zmi ' Zm1 m2
Zmi ' Zm1 m2 m3
Zmi ' · · · ' Zm1 m2 ···mn .
i=1
i=3
i=4
Q
Conversely, if i=1 Zmi ' Zm1 m2 ···mn , then ni=1 Zmi contains a subgroup
isomorphic
Q to Zmi × Zmj for 1 ≤ i < j ≤ n. Every subgroup of the cyclic
group ni=1 Zmi ' Zm1 m2 ···mn is cyclic. Hence Zmi × Zmj is cyclic. So
gcd(mi , mj ) = 1 for 1 ≤ i < j ≤ n.
Qn
Cor 2.52. 6 If a positive integer n is factorized as a product of powers of
distinct prime numbers:
n = (p1 )n1 (p2 )n2 · · · (pr )nr ,
then
Zn ' Z(p1 )n1 × Z(p2 )n2 × · · · × Z(pr )nr .
Note: Each Z(pi )ni can not be further decomposed into a product of
proper nontrivial subgroups.
Ex 2.53. (Z5 )2 Z52 . The group (Z5 )2 is not cyclic, but Z52 = Z25 is
cyclic.
Def 2.54. The least common multiple (abbreviated lcm) of positive integers
r1 , r2 , · · · , rn is the smallest positive integer that is a multiple of each ri for
i = 1, 2, · · · , n.
If M is a multiple of ri for i = 1, · · · , n, then M is a multiple of
lcm(r1 , · · · , rn ).
Qn
Thm 2.55. Let (a1 , a2 , · · · , an ) ∈ Q
i=1 Gi . If ai is of finite order ri in Gi ,
then the order of (a1 , a2 , · · · , an ) in ni=1 Gi is lcm(r1 , r2 , · · · , rn ).
Proof.
(a1 , a2 , · · · , an )k = (ak1 , ak2 , · · · , akn ).
So
m
m
(am
1 , a2 , · · · , an ) = (e1 , e2 , · · · , en )
if and only if m is a multiple of ri for i = 1, 2, · · · , n, if and only if m
is a multiple of lcm(r1 , r2 , · · · , rn ). The smallest such positive integer is
lcm(r1 , r2 , · · · , rn ). So the order of (a1 , a2 , · · · , an ) is lcm(r1 , r2 , · · · , rn ).
Ex 2.56. The order of (8, 4, 10) in Z12 × Z60 × Z24 .
6
2nd HW: 13, 15, 32, 47, 50
34
CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS
2.4.2
The Structure of Finitely Generated Abelian Groups
Thm 2.57 (Fundamental Theorem of Finitely Generated Abelian
Groups). Every finitely generated abelian group G is isomorphic to a direct
product of cyclic groups in the form
Z(p1 )r1 × Z(p2 )r2 × · · · × Z(pn )rn × Z × Z × · · · × Z
where the pi are primes, not necessarily distinct, and the ri are positive
integers. The direct product is unique except for possible rearrangement of
the factors; that is, the number (Betti number of G) of factor Z is unique
and the prime powers (pi )ri are unique.
The proof (omitted) is challenging.
In particular, every finite abelian group is isomorphic to a product of
finite cyclic groups. (Caution: finite group 6= finitely generated group)
Ex 2.58. Decompose the following three groups:
Z4 × Z12 × Z15 ' Z22 × (Z22 × Z3 ) × (Z3 × Z5 ) ' (Z22 )2 × (Z3 )2 × Z5 ,
Z3 × Z4 × Z60 ' Z3 × Z22 × (Z22 × Z3 × Z5 ) ' (Z22 )2 × (Z3 )2 × Z5 ,
Z20 × Z36 ' (Z22 × Z5 ) × (Z22 × Z32 ) ' (Z22 )2 × Z32 × Z5 .
Therefore, Z4 × Z12 × Z15 ∼
= Z3 × Z4 × Z60 Z20 × Z36 .
Ex 2.59 (Ex 11.13, p.109). Find all abelian groups, up to isomorphism,
of order 360.
2.4.3
Applications
Def 2.60. A group G is decomposable if it is isomorphic to a direct product
of two proper nontrivial subgroups. Otherwise G is indecomposable.
Thm 2.61. The finite indecomposable abelian groups are exactly the cyclic
groups with order a power of a prime.
Thm 2.62. If G is a finite abelian group, then G has a subgroup of order
m for every m that divides |G|.
Ex 2.63. If m is a square-free integer, then every abelian group of order m
is isomorphic to Zm and is cyclic.
2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS35
2.4.4
Homework, II-11, p.110-p.113
1st 1, 2, 14, 34, 46
2nd 13, 15, 32, 47, 50
3rd 18, 26, 36, 53
(opt) 29, 37, 38, 39, 52, 54