Exploring Cyclic Groups: Abstract Algebra Basics

Table of Contents

1. Introduction
2. What is Z6?
3. Cyclic Subgroups of Z6
   1. Cyclic Subgroup Generated by 1
   2. Cyclic Subgroup Generated by 2
   3. Cyclic Subgroup Generated by 3
   4. Cyclic Subgroup Generated by 4
   5. Cyclic Subgroup Generated by 5
4. Cyclicity of Z6
   1. Definition of a Cyclic Group
   2. Definition of a Generator
   3. Cyclicity of Z6
5. Example with Group S3
6. Cyclic Subgroups of S3
   1. Cyclic Subgroup Generated by Identity
   2. Cyclic Subgroup Generated by Flips
   3. Cyclic Subgroup Generated by Rotations
7. Cyclicity of S3
8. Cyclic Groups
   1. Cyclic Group Z7
   2. Cyclic Group Z
   3. Cyclic Group 2Z + 3Z
9. Abelian and Cyclic Groups
   1. Are All Cyclic Groups Abelian?
   2. Are All Abelian Groups Cyclic?

Cyclic Groups: Exploring Subgroups and Cyclicity

Cyclic groups are an essential concept in mathematics that study the properties of groups generated by a single element. In this article, we will dive into the intriguing world of cyclic groups and explore their subgroups. We will begin by examining the group Z6, which consists of the integers 0, 1, 2, 3, 4, and 5 under the binary operation of addition modulo 6.

What is Z6?

Z6 is a finite cyclic group that operates under addition modulo 6. To understand how this group functions, let's consider an example. Suppose we want to compute the sum of 4 and 5. Normally, we would say that 4 plus 5 equals 9. However, in Z6, 9 is equivalent to 3 due to the addition modulo 6. The group table for Z6 helps us confirm that 4 plus 5 is indeed 3.

Cyclic Subgroups of Z6

Now, let's explore the cyclic subgroups of Z6. A cyclic subgroup is a subgroup generated by a single element. We will examine the subgroups generated by each element of Z6.

Cyclic Subgroup Generated by 1

We start with the element 1 and observe how it generates a subgroup. When we add 1 repeatedly, we get the sequence 1, 2, 3, 4, 5, 0, 1, 2... We can see that the subsequence generated by 1 wraps around and ultimately leads us back to where we started. Thus, the subgroup generated by 1 consists of the elements 1, 2, 3, 4, 5, and 0.

Cyclic Subgroup Generated by 2

Next, we examine the subgroup generated by the element 2. Adding 2 successively yields the sequence 2, 4, 0, 2, 4, 0... Similar to the previous subgroup, this subgroup also returns to its starting point. Therefore, the subgroup generated by 2 in Z6 is 2, 4, 0.

Cyclic Subgroup Generated by 3

Continuing this pattern, let's explore the subgroup generated by 3. The sequence formed by repeatedly adding 3 is 3, 0, 3, 0... This subgroup demonstrates the same behavior as the previous ones. Consequently, the subgroup generated by 3 is 3 and 0.

Cyclic Subgroup Generated by 4

Now, let's consider the element 4 and its generated subgroup. Adding 4 consecutively gives us the sequence 4, 2, 0, 4, 2, 0... Just like the prior subgroups, this one exhibits the same cyclic properties. Hence, the subgroup generated by 4 is 4, 2, 0.

Cyclic Subgroup Generated by 5

Lastly, we examine the subgroup generated by 5. The sequence produced by repeatedly adding 5 is 5, 4, 3, 2, 1, 0, 5, 4... As expected, this subgroup exhibits the same cyclic behavior. Therefore, the subgroup generated by 5 is 5, 4, 3, 2, 1, and 0.

Cyclicity of Z6

After observing the cyclic subgroups of Z6, we can now delve into the concept of cyclicity for groups. A group G is said to be cyclic if it contains an element such that the cyclic subgroup generated by that element is equal to the entire group G. In simpler terms, a group is cyclic if there exists a single element that can generate the entire group through repeated application of the group operation.

Definition of a Cyclic Group

Formally, a group G is said to be cyclic if there exists an element a in G such that G equals the group generated by that element a. We can also define a generator of a group G as an element a in G that can generate G by taking powers of a, denoted as a to the power of n, where n is any integer.

Cyclicity of Z6

By applying the definition of a cyclic group, we can determine the cyclicity of Z6. Looking back at the cyclic subgroups we explored earlier, we observed that the elements 1 and 5 generated the entire group Z6. Therefore, Z6 is a cyclic group with generators 1 and 5. This aligns with our understanding of the cyclicity concept, as Z6 can indeed be generated by these two elements.

Example with Group S3

Now, let's move on to another example to deepen our understanding of cyclic groups. Consider the group S3, which represents the symmetry operations on an equilateral triangle. In this case, we have three rows: the identity (Row not), rotations (Row 1 and Row 2), and flips across a line of symmetry (Meus). The binary operation for S3 is composition of functions, where two operations are performed in succession.

Cyclic Subgroups of S3

Similar to Z6, we will explore the cyclic subgroups of S3 in order to analyze their properties. Let's start by examining the subgroups generated by each element.

Cyclic Subgroup Generated by Identity

The identity element, represented by Row not, generates its own subgroup. As the identity, this subgroup only contains the identity itself.

Cyclic Subgroup Generated by Flips

Moving on to the flips (Meus), we can observe that each flip operation generates its respective subgroup. For instance, the subgroup generated by Mu1 consists solely of the identity and the flip Mu1. Similarly, the subgroup generated by Mu2 contains the identity and Mu2. The same pattern holds for Mu3 and its generated subgroup.

Cyclic Subgroup Generated by Rotations

Lastly, let's examine the cyclic subgroups generated by the rotations (Row 1 and Row 2). Upon analyzing the group table, we can deduce that these rotations do not generate the entire group S3. Instead, they generate a smaller subgroup that consists of the identity and one rotation. For example, the subgroup generated by Row 1 contains the identity and Row 2. Similarly, the subgroup generated by Row 2 comprises the identity and Row 1.

Cyclicity of S3

Based on our analysis of the cyclic subgroups in S3, we can conclude that S3 is not a cyclic group. None of the single elements in S3 can generate the entire group, as opposed to Z6 where the elements 1 and 5 could generate the whole group. Thus, the absence of a single generator distinguishes S3 from being a cyclic group.

Cyclic Groups

So far, we have explored finite cyclic groups such as Z6 and S3. However, cyclic groups can be finite or infinite, as we will now explore a few examples.

Cyclic Group Z7

The group Z7 consists of the integers 0, 1, 2, 3, 4, 5, and 6, with the binary operation of addition modulo 7. Upon analyzing the subgroup generated by each element, we observe that all elements except 0 generate the entire group. In other words, Z7 is a cyclic group with generators 1 and 6.

Cyclic Group Z

The group Z, also known as the set of integers, operates under the binary operation of addition. Considering the subgroup generated by 1, we can see that 1 and -1 generate the entire group. Thus, Z is a cyclic group with generators 1 and -1.

Cyclic Group 2Z + 3Z

Lastly, let's examine the group 2Z + 3Z (read as 2Z plus 3Z). This group comprises elements that are multiples of 2 and multiples of 3. By analyzing the subgroup generated by 2, we find that 2 and 4 generate the entire group. Similarly, the elements 3 and 6 generate the entire group. Hence, 2Z + 3Z is a cyclic group with generators 2, 4, 3, and 6.

Abelian and Cyclic Groups

In closing, let's address the relationship between abelian and cyclic groups. An abelian group is a group in which the group operation is commutative. The question arises: are all cyclic groups abelian? Conversely, are all abelian groups cyclic?

Are All Cyclic Groups Abelian?

The answer is yes. In fact, all cyclic groups are abelian. This stems from the fact that the generators of a cyclic group commute with one another.

Are All Abelian Groups Cyclic?

However, not all abelian groups are cyclic. As we observed with S3, certain abelian groups do not possess a single element capable of generating the entire group. Therefore, while all cyclic groups are abelian, the converse does not hold true for all abelian groups.

In conclusion, cyclic groups offer an intriguing perspective on the concept of group generation. Through analyzing subgroups and exploring cyclicity, we gain a deeper understanding of these structures. Whether finite or infinite, cyclic groups play a significant role in the realm of mathematics and group theory