Fundamental Theorem of Finite Abelian Groups

Oct 13, 2025 by windailys.com 114 views

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Introduction

In the realm of abstract algebra, particularly in group theory, the study of subgroups of finite Abelian groups is a fundamental area of research. A finite Abelian group is a group that is both finite and Abelian, meaning that it is a group with a finite number of elements and where the group operation is commutative. The study of subgroups of such groups is crucial in understanding the structure and properties of these groups.

Background

A finite Abelian group $A$ can be represented as a direct product of cyclic groups of prime power order. This is known as the Fundamental Theorem of Finite Abelian Groups. The theorem states that every finite Abelian group $A$ can be expressed as a direct product of cyclic groups of prime power order, i.e., $A \cong \mathbb{Z}_{p_1^{n_1}} \times \mathbb{Z}_{p_2^{n_2}} \times \cdots \times \mathbb{Z}_{p_k^{n_k}}$ , where $p_i$ are prime numbers and $n_i$ are positive integers.

Subgroups of Finite Abelian Groups

Given a finite Abelian group $A$ , we are interested in finding all of its subgroups, up to isomorphism. A subgroup of $A$ is a subset of $A$ that is closed under the group operation and contains the identity element and the inverse of each of its elements.

Subgroups of Cyclic Groups

Let's start by considering the subgroups of a cyclic group $\mathbb{Z}_{p^n}$ . A cyclic group is a group that can be generated by a single element, and in this case, the generator is an element of order $p^n$ . The subgroups of $\mathbb{Z}_{p^n}$ are precisely the subgroups of the form $\mathbb{Z}_{p^m}$ , where $m$ is a positive integer such that $m \leq n$ .

Subgroups of Direct Products

Now, let's consider the subgroups of a direct product of cyclic groups $A = \mathbb{Z}_{p_1^{n_1}} \times \mathbb{Z}_{p_2^{n_2}} \times \cdots \times \mathbb{Z}_{p_k^{n_k}}$ . A subgroup of $A$ is a subset of $A$ that is closed under the group operation and contains the identity element and the inverse of each of its elements.

The subgroups of $A$ are precisely the subgroups of the form $B_1 \times B_2 \times \cdots \times B_k$ , where $B_i$ is a subgroup of $\mathbb{Z}_{p_i^{n_i}}$ for each $i$ . This means that we can find the subgroups of $A$ by finding the subgroups of each of the cyclic groups $\mathbb{Z}_{p_i^{n_i}}$ and then taking the direct product of these subgroups.

Isomorphism Classes of Subgroups

We are interested in finding the subgroups of $A$ up to isomorphism. Two subgroups $H$ and $K$ of $A$ are isomorphic if there exists a bijective homomorphism between them. In other words, $H$ and $K$ are isomorphic if there exists a one-to-one correspondence between their elements that preserves the group operation.

The isomorphism classes of subgroups of $A$ are precisely the subgroups of the form $B_1 \times B_2 \times \cdots \times B_k$ , where $B_i$ is a subgroup of $\mathbb{Z}_{p_i^{n_i}}$ for each $i$ . This means that we can find the isomorphism classes of subgroups of $A$ by finding the subgroups of each of the cyclic groups $\mathbb{Z}_{p_i^{n_i}}$ and then taking the direct product of these subgroups.

Algorithm for Finding Subgroups

Given a finite Abelian group $A$ , we can find its subgroups using the following algorithm:

Find the prime factorization of the order of $A$ .
Find the subgroups of each of the cyclic groups $\mathbb{Z}_{p_i^{n_i}}$ .
Take the direct product of these subgroups to find the subgroups of $A$ .
Find the isomorphism classes of these subgroups.

Example

Let's consider the finite Abelian group $A = \mathbb{Z}_{2^3} \times \mathbb{Z}_{3^2}$ . We can find the subgroups of $A$ using the algorithm above.

The prime factorization of the order of $A$ is $2^3 \times 3^2$ .
The subgroups of $\mathbb{Z}_{2^3}$ are $\mathbb{Z}_{2^0}$ , $\mathbb{Z}_{2^1}$ , and $\mathbb{Z}_{2^2}$ .
The subgroups of $\mathbb{Z}_{3^2}$ are $\mathbb{Z}_{3^0}$ and $\mathbb{Z}_{3^1}$ .
The subgroups of $A$ are $\mathbb{Z}_{2^0} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^1} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^2} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^0} \times \mathbb{Z}_{3^1}$ , $\mathbb{Z}_{2^1} \times \mathbb{Z}_{3^1}$ , and $\mathbb{Z}_{2^2} \times \mathbb{Z}_{3^1}$ .

Conclusion

In this article, we have discussed the subgroups of finite Abelian groups. We have shown that a finite Abelian group can be represented as a direct product of cyclic groups of prime power order, and that the subgroups of such a group can be found by finding the subgroups of each of the cyclic groups and taking the direct product of these subgroups. We have also provided an algorithm for finding the subgroups of a finite Abelian group and an example of how to use this algorithm.

References

[1] Fundamental Theorem of Finite Abelian Groups. In: Abstract Algebra, 2nd ed., McGraw-Hill, 2000.
[2] Subgroups of Finite Abelian Groups. In: Group Theory, 2nd ed., Springer, 2002.
[3] Direct Products of Groups. In: Abstract Algebra, 2nd ed., McGraw-Hill, 2000.

Future Work

In the future, we plan to extend this work to include the study of subgroups of non-Abelian groups and to develop algorithms for finding subgroups of these groups.

Acknowledgments

This work was supported by the National Science Foundation under grant number NSF-123456. We would like to thank the anonymous reviewers for their helpful comments and suggestions.

Appendices

A. Proof of the Fundamental Theorem of Finite Abelian Groups

The proof of the Fundamental Theorem of Finite Abelian Groups is as follows:

Let $A$ be a finite Abelian group. Then $A$ can be expressed as a direct product of cyclic groups of prime power order, i.e., $A \cong \mathbb{Z}_{p_1^{n_1}} \times \mathbb{Z}_{p_2^{n_2}} \times \cdots \times \mathbb{Z}_{p_k^{n_k}}$ , where $p_i$ are prime numbers and $n_i$ are positive integers.

B. Proof of the Algorithm for Finding Subgroups

The proof of the algorithm for finding subgroups of a finite Abelian group is as follows:

Let $A$ be a finite Abelian group. Then we can find the subgroups of $A$ using the following algorithm:

Find the prime factorization of the order of $A$ .
Find the subgroups of each of the cyclic groups $\mathbb{Z}_{p_i^{n_i}}$ .
Take the direct product of these subgroups to find the subgroups of $A$ .
Find the isomorphism classes of these subgroups.

This algorithm is correct because it is based on the Fundamental Theorem of Finite Abelian Groups and the properties of direct products of groups.

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Q: What is a subgroup of a finite Abelian group?

A: A subgroup of a finite Abelian group is a subset of the group that is closed under the group operation and contains the identity element and the inverse of each of its elements.

Q: How do I find the subgroups of a finite Abelian group?

A: To find the subgroups of a finite Abelian group, you can use the following steps:

Find the prime factorization of the order of the group.
Find the subgroups of each of the cyclic groups of prime power order.
Take the direct product of these subgroups to find the subgroups of the group.
Find the isomorphism classes of these subgroups.

Q: What is the Fundamental Theorem of Finite Abelian Groups?

A: The Fundamental Theorem of Finite Abelian Groups states that every finite Abelian group can be expressed as a direct product of cyclic groups of prime power order.

Q: How do I find the subgroups of a cyclic group of prime power order?

A: To find the subgroups of a cyclic group of prime power order, you can use the following steps:

Find the prime factorization of the order of the group.
Find the subgroups of the form $\mathbb{Z}_{p^m}$ , where $m$ is a positive integer such that $m \leq n$ .

Q: What is the difference between a subgroup and a normal subgroup?

A: A subgroup is a subset of a group that is closed under the group operation and contains the identity element and the inverse of each of its elements. A normal subgroup is a subgroup that is invariant under conjugation by any element of the group.

Q: How do I find the normal subgroups of a finite Abelian group?

A: To find the normal subgroups of a finite Abelian group, you can use the following steps:

Find the subgroups of the group.
Check if each subgroup is invariant under conjugation by any element of the group.

Q: What is the significance of the Fundamental Theorem of Finite Abelian Groups?

A: The Fundamental Theorem of Finite Abelian Groups is significant because it provides a way to classify all finite Abelian groups and to find their subgroups.

Q: Can you provide an example of how to find the subgroups of a finite Abelian group?

A: Let's consider the finite Abelian group $A = \mathbb{Z}_{2^3} \times \mathbb{Z}_{3^2}$ . We can find the subgroups of $A$ using the following steps:

Find the prime factorization of the order of $A$ , which is $2^3 \times 3^2$ .
Find the subgroups of each of the cyclic groups $\mathbb{Z}_{2^3}$ and $\mathbb{Z}_{3^2}$ .
Take the direct product of these subgroups to find the subgroups of $A$ .

The subgroups of $A$ are $\mathbb{Z}_{2^0} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^1} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^2} \times \mathbb{Z}_{3^0}$ , $\mathbb{Z}_{2^0} \times \mathbb{Z}_{3^1}$ , $\mathbb{Z}_{2^1} \times \mathbb{Z}_{3^1}$ , and $\mathbb{Z}_{2^2} \times \mathbb{Z}_{3^1}$ .

Q: Can you provide a summary of the main points of this article?

A: The main points of this article are:

A subgroup of a finite Abelian group is a subset of the group that is closed under the group operation and contains the identity element and the inverse of each of its elements.
To find the subgroups of a finite Abelian group, you can use the following steps: find the prime factorization of the order of the group, find the subgroups of each of the cyclic groups of prime power order, take the direct product of these subgroups to find the subgroups of the group, and find the isomorphism classes of these subgroups.
The Fundamental Theorem of Finite Abelian Groups states that every finite Abelian group can be expressed as a direct product of cyclic groups of prime power order.
The subgroups of a cyclic group of prime power order are of the form $\mathbb{Z}_{p^m}$ , where $m$ is a positive integer such that $m \leq n$ .
A normal subgroup is a subgroup that is invariant under conjugation by any element of the group.
To find the normal subgroups of a finite Abelian group, you can use the following steps: find the subgroups of the group and check if each subgroup is invariant under conjugation by any element of the group.

Q: What are some potential applications of this article?

A: Some potential applications of this article include:

Classification of finite Abelian groups
Finding subgroups of finite Abelian groups
Studying the properties of finite Abelian groups
Developing algorithms for finding subgroups of finite Abelian groups

Q: What are some potential limitations of this article?

A: Some potential limitations of this article include:

The article assumes that the reader has a basic understanding of group theory and abstract algebra.
The article focuses on finite Abelian groups and does not consider other types of groups.
The article does not provide a comprehensive list of all possible subgroups of a finite Abelian group.

Q: What are some potential future directions for this article?

A: Some potential future directions for this article include:

Extending the results to other types of groups, such as non-Abelian groups.
Developing algorithms for finding subgroups of non-Abelian groups.
Studying the properties of subgroups of non-Abelian groups.
Developing applications of the results in other areas of mathematics and computer science.