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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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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.

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*, â€¦., *gn* in *G*, such that every *g* in *G* can be written in this form:

*g* = *a1**g1* + *a2**g2* +...+ *an**gn*

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.

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:

- The number of terms in the product
- 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.

Since 8 = 4 x 2, we know that one of the finitely generated abelian groups of size 8 will be identical to the group *Z4* X *Z2*. By the way, this is also identical to *Z2* X *Z4*, since finitely generated abelian groups are commutative, so we only count this direct product once. Note here that the factors 4 and 2 determined the size of each cyclic group we included.

Since 8 = 2 x 2 x 2, we also know that another finite group of size 8 will be identical to the group *Z2* X *Z2* X *Z2*. Again, note that the order of the cyclic groups correspond to the factorization 2 x 2 x 2 of 8.

For larger group sizes, which can have more possible factorizations, you can follow the same process to find the possible direct products of cyclic groups. Just make sure to not double count cyclic group products that are identical to each other through commutativity.

Here are some examples of finitely generated abelian groups:

1. The integers together with addition are a finitely generated abelian group, since any set that includes 1 or -1 will generate it. To see this, note that you can obtain any integer by just adding together the appropriate number of 1's. For example, you can obtain the integer 4 by adding 1 + 1 + 1 + 1. Now since *Z* is a group, you also automatically have the identity element 0 for addition, as well as the inverse of all positive numbers. So, if you were able to generate 4 using just 1's, then you would also have -4 as it's the inverse element of 4.

2. Any cyclic group, such as *Z3*, is also a finitely generated abelian group. Recall that *Z3* is a group together with the operation addition modulo 3. This modulo operation means that any element of this group can be expressed as the possible remainder you get (0, 1, or 2) when dividing any number by 3. For example we could express 5 in this group as 5 â‰¡ 2 mod 3 (since 5 divided by 3 gives a remainder of 2). Let's see how we'd generate *Z3* with {2} as our finite set:

2 â‰¡ 2 mod 3

2 + 2 â‰¡ 4 â‰¡ 1 mod 3

2 + 2 + 2 â‰¡ 6 â‰¡ 0 mod 3

Hence, we were able to obtain the entire group *Z3* = {0, 1, 2} just by adding 2 to itself a certain amount of times.

Recall that a **finitely generated abelian group** is an abelian group, *G*, that can be generated from a finite set {*g1*, *g2*, â€¦., *gn*}, such that every *g* in *G* can be written in this form:

*g* = *a1**g1* + *a2**g2* +...+ *an**gn* where *a1*,..., *an* are integers.

According to the **fundamental theorem of finitely generated abelian groups**, they can be classified based on how they can be broken up into a direct product of cyclic groups *Zf1*,..., *Zfk*.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

- Number Theory: Divisibility & Division Algorithm 6:52
- Euclidean Algorithm & Diophantine Equation: Examples & Solutions 7:01
- Fermat's Last Theorem: Definition & Example 5:10
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