Classifying Finite Abelian Groups

In summary, the conversation discusses the process of counting and describing the different isomorphism classes of abelian groups of order 1800. The speaker is struggling with using the classification theorem and the Sylow theorems to determine all the possibilities. The other person suggests using only the classification theorem, which should provide enough information to identify subgroups of order 8, 9, and 25, among others.
  • #1
Kalinka35
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Homework Statement


Count and describe the different isomorphism classes of abelian groups of order 1800. I don't need to list the group individually, but I need to give some sort of justification.

Homework Equations


The Attempt at a Solution


I'm using the theorem to classify finitely generated abelian groups,
As always we will have Z_1800 to begin with.
Also we know 1800=23(32)(52).

But how do I count all of the possibilities?
 
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  • #2
Why don't you start listing them individually? Maybe that will jog your imagination enough to figure out how to count them without listing them.
 
  • #3
I've counted all the ones that you get by using the classification theorem for finite abelian groups but a problem I've encountered is using the information given by the Sylow theorems as none of the groups from the first theorem contain the necessary subgroups.
What I've gathered is that there must be at least one subgroup of order 8, 9, and 25 and there a bunch of possibilities for how many for all of them. The numbers just don't really add up right. Any ideas how to approach this?
 
  • #4
You don't need the Sylow theorems. You have a complete classification theorem for abelian groups. That's all you need. That certainly gives you subgroups of order 8, 9 and 25. And several other orders besides. I'm not sure what you are fretting about. Which structure from the classification theorem doesn't give you these?
 

FAQ: Classifying Finite Abelian Groups

1. What are finite abelian groups?

Finite abelian groups are a type of mathematical structure that consists of a finite set of elements and a binary operation (usually denoted as +) that follows the commutative and associative properties. This means that the order in which the group elements are combined does not affect the outcome of the operation. Additionally, each element in the group has an inverse, meaning that when combined with another element, it results in the identity element (usually denoted as 0 or e).

2. How do you classify finite abelian groups?

Finite abelian groups can be classified by their order (number of elements) and the number of cyclic subgroups they contain. This can be done using the fundamental theorem of finite abelian groups, which states that any finite abelian group can be written as a direct product of cyclic groups of prime power order.

3. What is a cyclic subgroup?

A cyclic subgroup is a subgroup of a group that is generated by a single element. This means that by repeatedly combining the element with itself using the group operation, all other elements in the subgroup can be obtained. In finite abelian groups, the number of cyclic subgroups is an important factor in their classification.

4. What is the significance of classifying finite abelian groups?

Classifying finite abelian groups allows us to better understand their properties and relationships with other mathematical structures. It also helps in solving problems and making predictions about group behavior. Additionally, this classification is used in various areas of mathematics, such as cryptography and coding theory.

5. Can finite abelian groups be infinitely large?

No, by definition, finite abelian groups have a finite number of elements. However, there are infinite abelian groups, such as the integers under addition, which share some properties with finite abelian groups but have infinitely many elements. The classification and properties of infinite abelian groups are more complex and fall under the study of abstract algebra.

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