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Finitely Generated Abelian Groups: Classification & Examples

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  • 0:04 Finitely Generated…
  • 2:27 Classification
  • 4:51 Examples
  • 6:24 Lesson Summary
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Lesson Transcript
Instructor: Laura Golnabi

Laura Golnabi is a Ph.D. student in Mathematics Education at Teachers College, Columbia University. She also teaches undergraduate mathematics courses, and has developed problem solving courses designed for non-STEM majors. Her current research involves if and when students suffering from mathematics anxiety are able to have positive, flow-like experiences in mathematics.

In this lesson, you'll explore the definition of finitely generated abelian groups and how they can be classified. You'll also be introduced to some examples and their classification.

Finitely Generated Abelian Groups

Before jumping into the definition of a finitely generated abelian group, let's unpack this term a little and take it word by word. By now, you may already be familiar with the definition of a group, but just in case, a group is a set together with an operation that meets the following set of rules:

  • Closure: For any a and b in the group, the result of a * b should also be in the group.
  • Associativity: For any a, b, and c in the group, the operation * (multiplication) must be associative. That is, we must have the following:
    • (a * b) * c = a * (b * c)
  • Identity: There must exist an identity element e in the group that satisfies this requirement for all other elements a in the group:
    • a * e = a and e * a = a
  • Inverse: For every a in the group, there must exist another element b in the group such that satisfies this relation:
    • a * b = e and b * a = e

Next, the word abelian is just a fancy way of saying that the group is commutative. In other words, an abelian group is just a group with one extra rule to follow:

  • Commutativity: For every a and b in the group, the following must hold:
    • a * b = b * a

In this context, adding the word finitely generated tells us that the abelian group is generated from a finite set, or a set with a limited number of elements. For example, we could use the finite set {g1, ...gn}, which has n elements, to generate an abelian group G.

Hence, a finitely generated abelian group is an abelian group, G, for which there exists finitely many elements g1, g2, …., gn in G, such that every g in G can be written in this form:

g = a1g1 + a2g2 +...+ angn

where a1,..., an are integers

Note that being finitely generated does not mean that the group itself must be finite. It only means that there exists a finite set that can generate the group. Consider the set of integers Z together with the regular addition operation. This group itself is infinite (you can always keep adding 1), but since a finite set such as {1} can generate it, it's considered finitely generated.

Classification

The fundamental theorem of finitely generated abelian groups tells us that every finitely generated abelian group is isomorphic to a direct product of cyclic groups. Cyclic groups are groups that can be generated by just one element. You may see versions of this theorem that use direct sum instead of direct product, but these operations mean the same thing for abelian groups in this context.

In particular, finitely generated abelian groups are determined by two things:

  1. The number of terms in the product
  2. The size of each cyclic group, which is also known as the order of the cyclic group

In other words, we can classify finitely generated abelian groups based on how they can be broken up into a product of cyclic groups Zf1,..., Zfk. Note that f1,..., fk are prime factors raised to a positive integer of 1 or greater. They make up a factorization of the size of the group, and each group Zfi is the cyclic group of order fi.

That is, you would begin by taking different factorizations of the order (size) of the finitely generated abelian group and create products of cyclic groups based on each factorization.

That sounds like a lot, but let's look at a quick example to see how this works. Say we have a finitely generated abelian group of size 8, then, depending on how you break up the factors of 8, there are two possible ways of classifying them.

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