Lecture 4.4: Finitely generated abelian groups Matthew Macauley - PowerPoint PPT Presentation

Lecture 4.4: Finitely generated abelian groups Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 1 / 7

Finite abelian groups We’ve seen that some cyclic groups can be expressed as a direct product, and others cannot. Below are two ways to lay out the Cayley diagram of Z 6 so the direct product structure is obvious: Z 6 ∼ = Z 3 × Z 2 . 0 0 2 4 3 1 5 4 2 3 5 1 However, the group Z 8 cannot be written as a direct product. No matter how we draw the Cayley graph, there must be an element (arrow) of order 8. (Why?) We will answer the question of when Z n × Z m ∼ = Z nm , and in doing so, completely classify all finite abelian groups. M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 2 / 7

Finite abelian groups Proposition Z nm ∼ = Z n × Z m if and only if gcd( n , m ) = 1. Proof (sketch) “ ⇐ ”: Suppose gcd( n , m ) = 1. We claim that (1 , 1) ∈ Z n × Z m has order nm . | (1 , 1) | is the smallest k such that “( k , k ) = (0 , 0).” This happens iff n | k and m | k . Thus, k = lcm( n , m ) = nm . � (0 , 0) (1 , 0) (2 , 0) (3 , 0) The following image illustrates this using the (0 , 1) (1 , 1) (2 , 1) (3 , 1) Cayley diagram in the group Z 4 × Z 3 ∼ = Z 12 . · · · (0 , 2) (1 , 2) (2 , 2) (3 , 2) M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 3 / 7

Finite abelian groups Proposition Z nm ∼ = Z n × Z m if an only if gcd( n , m ) = 1. Proof (cont.) “ ⇒ ”: Suppose Z nm ∼ = Z n × Z m . Then Z n × Z m has an element ( a , b ) of order nm . For convenience, we will switch to “multiplicative notation”, and denote our m -1) cyclic groups by C n . ( e , e ) ( e , b ) ( e , b . . . Clearly, � a � = C n and � b � = C m . Let’s look at . . . m -1) a Cayley diagram for C n × C m . ( a , e ) ( a , b ) ( a , b The order of ( a , b ) must be a multiple of n . . . ... . . . (the number of rows), and of m (the number . . . of columns). . . . ( an -1 , e ) ( an -1 , b ) n -1 m -1 By definition, this is the least common a , b multiple of n and m . But | ( a , b ) | = nm , and so lcm( n , m ) = nm . Therefore, gcd( n , m ) = 1. � M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 4 / 7

The Fundamental Theorem of Finite Abelian Groups Classification theorem (by “prime powers”) Every finite abelian group A is isomorphic to a direct product of cyclic groups, i.e., for some integers n 1 , n 2 , . . . , n m , A ∼ = Z n 1 × Z n 2 × · · · × Z n m , where each n i is a prime power, i.e., n i = p d i i , where p i is prime and d i ∈ N . The proof of this is more advanced, and while it is at the undergraduate level, we don’t yet have the tools to do it. However, we will be more interested in understanding and utilizing this result. Example Up to isomorphism, there are 6 abelian groups of order 200 = 2 3 · 5 2 : Z 8 × Z 25 Z 8 × Z 5 × Z 5 Z 2 × Z 4 × Z 25 Z 2 × Z 4 × Z 5 × Z 5 Z 2 × Z 2 × Z 2 × Z 25 Z 2 × Z 2 × Z 2 × Z 5 × Z 5 M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 5 / 7

The Fundamental Theorem of Finite Abelian Groups Finite abelian groups can be classified by their “elementary divisors.” The mysterious terminology comes from the theory of modules (a graduate-level topic). Classification theorem (by “elementary divisors”) Every finite abelian group A is isomorphic to a direct product of cyclic groups, i.e., for some integers k 1 , k 2 , . . . , k m , A ∼ = Z k 1 × Z k 2 × · · · × Z k m . where each k i is a multiple of k i +1 . Example Up to isomorphism, there are 6 abelian groups of order 200 = 2 3 · 5 2 : by “prime-powers” by “elementary divisors” Z 8 × Z 25 Z 200 Z 4 × Z 2 × Z 25 Z 100 × Z 2 Z 2 × Z 2 × Z 2 × Z 25 Z 50 × Z 2 × Z 2 Z 8 × Z 5 × Z 5 Z 40 × Z 5 Z 4 × Z 2 × Z 5 × Z 5 Z 20 × Z 10 Z 2 × Z 2 × Z 2 × Z 5 × Z 5 Z 10 × Z 10 × Z 2 M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 6 / 7

The Fundamental Theorem of Finitely Generated Abelian Groups Just for fun, here is the classification theorem for all finitely generated abelian groups. Note that it is not much different. Theorem Every finitely generated abelian group A is isomorphic to a direct product of cyclic groups, i.e., for some integers n 1 , n 2 , . . . , n m , A ∼ = Z × · · · × Z × Z n 1 × Z n 2 × · · · × Z n m , � �� � k copies where each n i is a prime power, i.e., n i = p d i i , where p i is prime and d i ∈ N . In other words, A is isomorphic to a (multiplicative) group with presentation: A = � a 1 , . . . , a k , r 1 , . . . , r m | r n i = 1 , a i a j = a j a i , r i r j = r j r i , a i r j = r j a i � . i In summary, (finitely generated) abelian groups are relatively easy to understand. In contrast, nonabelian groups are more mysterious and complicated. Soon, we will study the Sylow Theorems which will help us better understand the structure of finite nonabelian groups. M. Macauley (Clemson) Lecture 4.4: Finitely generated abelian groups Math 4120, Modern Algebra 7 / 7