Section 11 Direct Products and Finitely Generated Abelian

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 S 1, S 2, …, Sn is the set of all ordered ntuples (a 1, a 2, …, an), where ai Si for i=1, 2, …, n. The Cartesian product is denoted by either S 1 S 2 … Sn or by

Theorem Let G 1, G 2, …, Gn be groups. For (a 1, a 2, …, an) and (b 1, b 2, …, bn) in , Define (a 1, a 2, …, an)(b 1, b 2, …, bn) to be the element (a 1 b 1, a 2 b 2, …, an bn). Then is a group, the direct product of the groups Gi, under this binary operation. Proof: exercise. Note: • In the event that the operation of each Gi is commutative, we sometimes use additive notation in and refer to as the direct sum of the groups Gi. • If the Si has ri elements for i=1, …, n, then has r 1 r 2, …, rn elements.

Example: Determine if Z 2 Z 3 is cyclic. Solution: | Z 2 Z 3 |=6 and Z 2 Z 3 ={(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)}. Here the operations in Z 2, , Z 3 are written additively. We can check that (1, 1) is the generator, so Z 2 Z 3 is cyclic. Hence Z 2 Z 3 is isomorphic to Z 6. (there is, up to isomorphism, only one cyclic group structure of a given order. ) Example: Determine if Z 3 is cyclic. Solution: We claim Z 3 is not cyclic. |Z 3 Z 3|=9, but every element in Z 3 can only generate three elements. So there is no generator for Z 3. Hence Z 3 is not isomorphic to Z 9. Similarly, Z 2 is not cyclic, Thus Z 2 must be isomorphic to Z 6.

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 The group is cyclic and isomorphic to Zm 1 m 2. . mn if and only if the 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 Then Zn is isomorphic to Example: Z 72 is isomorphic to Z 8 Z 9.

Least Common Multiple Definition Let r 1 r 2, …, 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 r 1 r 2, …, 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 Let (a 1, a 2, …, an) . If ai is of finite order ri in Gi, then the order of (a 1, a 2, …, an) in is equal to the least common multiple of all the ri.

Example: Find the order of (8, 4, 10) in the group Z 12 Z 60 Z 24. Solution: The order of 8 in Z 12 is 12/gcd(8, 12)=3, the order of 4 in Z 60 is 60/gcd(4, 60)=15, and the order of 10 in Z 24 is 24/gcd(10, 24)=12. The lcm(3, 5, 12)=60, so (8, 4, 10) is or order 60 in the group Z 12 Z 60 Z 24.

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 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 are unique.

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 theorem. 1. 2. 3. 4. 5. 6. Since 360=23325. Then by theorem, we get the following: Z 2 Z 3 Z 5 Z 2 Z 4 Z 3 Z 5 Z 2 Z 9 Z 5 Z 2 Z 4 Z 9 Z 5 Z 8 Z 3 Z 5 Z 8 Z 9 Z 5 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.