Transcript Abstract Algebra

Section 11 Direct Products
and Finitely Generated Abelian Groups
One purpose of this section is to show a way to use known groups as
building blocks to form more groups.
Definition:
The Cartesian product of sets S1, S2, …,Sn is the set of all ordered ntuples (a1, a2, …,an), where aiSi for i=1, 2, …, n. The Cartesian
product is denoted by either
S1S2 …Sn
or by
n
 Si
i 1
Theorem
Theorem
n
Let G1, G2, …,Gn be groups. For (a1, a2, …,an) and (b1, b2, …,bn) in  Gi ,
i 1
Define (a1,n a2, …,an)(b1, b2, …,bn) to be the element (a1 b1, a2 b2, …,an bn).
Then  Gi is a group, the direct product of the groups Gi, under this
i 1
binary operation.
Proof: exercise.
Note:
• In the event that the operation of each
Gi is commutative,
we
n
n
sometimes use additive notation in  Gi and refer to  Gi as the
i 1
i 1
direct sum of the groups Gi.
n
Si has r r ,…,r elements.
• If the Si has ri elements for i=1, …,n, then 
1 2
n
i 1
Example
Example: Determine if Z2  Z3 is cyclic.
Solution: | Z2  Z3 |=6 and Z2  Z3 ={(0, 0),(0, 1),(0, 2),(1, 0),(1, 1),(1, 2)}.
Here the operations in Z2, , Z3 are written additively.
We can check that (1, 1) is the generator, so Z2  Z3 is cyclic.
Hence Z2  Z3 is isomorphic to Z6.
(there is, up to isomorphism, only one cyclic group structure of a given order.)
Example: Determine if Z3  Z3 is cyclic.
Solution: We claim Z3  Z3 is not cyclic. |Z3  Z3|=9, but every element in
Z3  Z3 can only generate three elements. So there is no generator for
Z3  Z3. Hence Z3  Z3 is not isomorphic to Z9.
Similarly, Z2  Z2 is not cyclic, Thus Z2  Z2 must be isomorphic to Z6.
Theorem
Theorem
The group ZmZn is cyclic and is isomorphic to Zmn if and only if m and
n are relatively prime, that is, the gcd of m and n is 1.
Corollary
n
The group  Z mi is cyclic and isomorphic to Zm1m2..mn if and only if the
i 1
numbers for i =1, …, n are such that the gcd of any two of them is
1.
Example
The previous corollary shows that if n is written as a product of powers
of distinct prime numbers, as in
n  ( p1 ) n1 ( p2 ) n2  ( pr ) nr
Then Zn is isomorphic to
Z( p )n1  Z( p )n2 Z( p )nr
1
2
Example: Z72 is isomorphic to Z8  Z9.
r
Least Common Multiple
Definition
Let r1r2,…,rn be positive integers. Their least common multiple (lcm) is
the positive integer of the cyclic group of all common multiples of the
ri, that is, the cyclic group of all integers divisible by each ri for i=1, 2,
…, n.
Note: from the definition and the work on cyclic groups, we see that the
lcm of r1r2,…,rn is the smallest positive integer that is a multiple of
each ri for i=1, 2, …, n, hence the name least common multiple.
Theorem
Theorem
n
Let (a1, a2, …,an) Gi . If ai is of finite order ri in Gi, then the order of (a1,
i 1
n
a2, …,an) in  Gi is equal to the least common multiple of all the ri.
i 1
Example
Example: Find the order of (8, 4, 10) in the group Z12  Z60  Z24.
Solution: The order of 8 in Z12 is 12/gcd(8, 12)=3,
the order of 4 in Z60 is 60/gcd(4, 60)=15, and
the order of 10 in Z24 is 24/gcd(10, 24)=12.
The lcm(3, 5, 12)=60, so (8, 4, 10) is or order 60 in the
group Z12  Z60  Z24.
The structure of Finitely Generated Abelian Groups
Theorem (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 ( p )r1  Z ( p
1
2)
r2
 Z ( p
n)
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 factors Z is unique
and the prime powers ( pi ) ri are unique.
Example
Example: Find all abelian groups, up to isomorphism, of order 360.
Solution: Since the groups are to be of the finite order 360, no factors Z
will appear in the direct product in the theorem.
1.
2.
3.
4.
5.
6.
Since 360=23325. Then by theorem, we get the following:
Z2  Z2  Z2  Z3  Z3  Z5
Z2  Z4  Z3  Z3  Z5
Z2  Z2  Z2  Z9  Z5
Z2  Z4  Z9  Z5
Z8  Z3  Z3  Z5
Z8  Z9  Z5
There are six different abelian groups (up to isomorphism) of order 360.
Application
Definition
A group G is decomposable if it is isomorphic to a direct product of two
proper nontrivial subgroups. Otherwise G is indecomposable.
Theorem
The finite indecomposable abelian groups are exactly the cyclic groups
with order a power of a prime.
Theorem
If m divides the order of a finite abelian group G, then G has a subgroup of
order m.
Theorem
If m is a square free integer, that is, m is not divisible of the square of any
prime, then every abelian group of order m is cyclic.