Lesson 35: Like Riding A Bicycle

In the previous lesson we revisited modular arithmetic and got more familiar with “clock addition.” We saw that we can define addition on sets like $\{0, 1, 2, 3, 4\}$ or $\{0, 1, 2\}$ or $\{0, 1, 2, 3, ..., N\}$ for any positive whole number $N,$ so long was we “cycle back through” to zero once we pass the highest number in the set.

In this lesson we’ll introduce a type of group known as a cyclic group (hence the title of this lesson). As we will see, the groups with modular arithmetic that we discussed last lesson will be examples of cyclic groups. In fact, the very structure that motivated our definition of a group in the first place (namely, the integers with the abstract group multiplication operation being regular addition) turns out to be a cyclic group as well. Indeed, we have seen many cyclic groups already, and our identification of all of them as cyclic groups will help us to see the similarities between seemingly wildly different groups.

Before diving in to the definition of a cyclic group and looking at examples, let us take a moment to, as usual, reflect on why we even want to make such definitions. Recall that this entire process (way back in lesson 2) got kicked off by us wanting to define the simplest, most fundamental objects that we could think of and then build mathematics up from that. We therefore defined elements and sets, then we saw that we could define functions between them. After studying some examples of functions we saw that some functions have special properties, and so we gave those special functions names so that we can distinguish them from the less special and more general functions. We then later saw that some sets have special properties (like being able to combine elements in certain ways (like addition or multiplication or whatever else) to get other elements), and so we decided to call these sets groups in order to distinguish them from the less special and more general sets.

After studying some examples of groups we now find ourselves in the similar situation of seeing that some groups have special properties, and so we want to introduce a name for these more special groups in order to distinguish them from the less special and more general ones. This process will continue on no matter which field of math we’re studying (i.e., no matter what kinds of mathematical structures we’re considering). Indeed, by continually studying structures with more and different special properties, we will end up seeing connections between structures that initially seemed wildly different, and these connections are some of the most beautiful truths that the human intellect has the ability to access. To get there, though, we have a lot of work to do, so let us now actually get on to studying cyclic groups.

Let us begin by making a simple observation. Let us recall from Lesson 29 that we can consider the group of rotations and reflections of a square napkin. This group, which we called the dihedral group of the square and denoted by $D_4$ consisted of 8 elements. Namely, we have

$D_4=\{e, \rho_1, \rho_2, \rho_3, r_1, r_2, r_3, r_4\}$

where $e$ is the identity element corresponding to “making no rotation or reflection at all,” the three elements $\rho_1, \rho_2, \rho_3$ corresponding, respectively, to rotation by 90 degrees, 180 degrees, and 270 degrees, and the elements $r_1, r_2, r_3, r_4$ corresponding to four different reflections of the square. In Lesson 29 we noted that the set consisting of the identity element and all three rotations—namely, the set $\{e, \rho_1, \rho_2, \rho_3\}$ —forms a subgroup of $D_4.$ This is simply because the composition of any two rotations gives another rotation.

Let us review how the rotations in this subgroup add together. We recall that a rotation by 360 degrees is precisely the same as doing nothing at all, so that $\rho_1\cdot \rho_3=e$ and $\rho_2\cdot\rho_2=e.$ In words, these two equations mean the following: “First rotating by 270 degrees and then rotating by 90 degrees is equal to doing nothing,” and “First rotating by 180 degrees and then rotating by 180 degrees is equal to doing nothing.” Both of these statements are definitely true, since 270 plus 90 is 360, as is 180 plus 180. Similarly, we have that $\rho_2\cdot \rho_3=\rho_1,$ as this says that rotating by 270 degrees and then by 180 degrees is identical to rotating by 90 degrees (remember that when we write $\rho_2\cdot \rho_3,$ we mean that we first do the rotation on the right first, and then the one on the left, so that we read this product from right to left (also remember that all rotations are in the same direction (and counterclockwise) so that a counterclockwise rotation is followed by another counterclockwise rotation)).

What we notice is that we have a similar “cycling back through” property going on here: we consider a 360 degree rotation as equivalent to a 0 degree rotation (i.e., doing nothing), and so we simply cycle back through once we hit 360 degrees and keep adding back up like normal. Indeed, we notice that the elements in $\{e, \rho_1, \rho_2, \rho_3\}$ add in exactly the same way as the elements in $\{0, 1, 2, 3\}$ if we simply “identify” the number 0 with $e,$ the number 1 with $\rho_1,$ the number 2 with $\rho_2,$ and the number 3 with $\rho_3.$ (Here, we keep “identify” in quotes because this term doesn’t have any rigor yet, i.e., we haven’t discussed what we precisely mean by “identify,” but we will do so in a few short lessons when we learn about isomorphisms.)

So what is it about the group $\{e, \rho_1, \rho_2, \rho_3\}$ that makes it so similar to the group $\{0, 1, 2, 3\}$ ? Indeed, there are many things that make these groups similar, for we will soon see that we can effectively treat these groups as being identical. For now, though, we will focus on only one aspect of these groups that make them similar. As a hint, the property that we will focus on will also be shared by the (infinite) group of integers

$\mathbb{Z}=\{..., -3, -2, -1, 0, 1, 2, 3, ...\}.$

So what is it that is similar between the groups $\mathbb{Z}=\{..., -3, -2, -1, 0, 1, 2, 3, ...\},$ the modular arithmetic groups like $\{0, 1, 2, 3, 4\}$ and $\{0, 1, 2, 3\},$ and the groups of rotations like $\{e, \rho_1, \rho_2, \rho_3\}$ or $\{e, \rho_1, \rho_2, \rho_3, \rho_4, \rho_5\}$ (where this last group is the group of rotations of a regular six-sided polygon, namely, a regular hexagon, where $\rho_1$ is a rotation by 60 degrees, $\rho_2$ is a rotation by 120 degrees, and so on).

Let’s take a second to really think about this. The similarities between the groups of rotations and the modular arithmetic groups might be obvious (or they might not be), because in some sense (which we’ll make precise in a couple of lessons) they’re basically identical. However, we also want to view these groups as somehow similar to the entire group of integers $\{..., -3, -2, -1, 0, 1, 2, 3, ...\},$ as well, and this similarity is a bit less obvious.

As a hint, let us see what groups these groups are not similar to. In particular, we notice that there is something very different between these groups and, say, the full dihedral group of the square $D_4.$ Additionally, we see that the groups mentioned in the previous two paragraphs are also somehow fundamentally different from the groups of permutations that we studied in lessons 24, 25, and 28. In particular, what we’re seeing is that the rotation groups, modular arithmetic groups, and the group of integers all kind of “go in one direction,” whereas the full dihedral group and the groups of permutations don’t have any kind of “directionality.” For example, the rotations and reflections get all mixed up when we multiply them in the dihedral groups: rotations multiplied by rotations give back rotations, whereas rotations multiplied by reflections give reflections, and reflections multiplied by reflections give rotations back!

But what do we really mean by “go in one direction” here? After giving it some thought, we find that what we really mean is that in the case of the integers, the modular arithmetic groups, and the rotation groups, there is an element that we can just multiply to itself (remember, when it comes to groups we always mean “abstract group multiply” when we say “multiply,” so that “multiplying” in the group of integers actually just means adding!) over and over again to get all of the other elements in the group.

For example, in the group $\{e, \rho_1, \rho_2, \rho_3\}$ we see that we can start with $\rho_1$ and multiply it to itself to give $\rho_2.$ Namely, we have $\rho_1\cdot \rho_1=\rho_2$ since rotating by 90 degrees and then by 90 degrees again gives a rotation by 180 degrees. We can then multiply this result by $\rho_1$ again to get $\rho_3,$ and then multiply this again to get the identity element $e,$ at which point multiplying by $\rho_1$ again simply gives $\rho_1$ back again, so that we’re back to where we started. Thus, we see that any element in the set $\{e, \rho_1, \rho_2, \rho_3\}$ can be written as $\rho_3^m$ for some number $m,$ where the notation $\rho_1^m$ simply means, in words, “multiply $\rho_1$ to itself $m$ times.” Thus, we see that $\rho_3=\rho_1^3$ because we get $\rho_3$ by multiplying $\rho_1$ to itself $3$ times.

Similarly, in the modular addition groups, each element in the group can be obtained by “multiplying” (remember, this means adding!) $1$ to itself over and over again. For example, in the set $\{0, 1, 2, 3, 4\},$ we see that $4=1+1+1+1$ so that we can write $4=1^4$ (where we remember that $1^4$ means “multiply $1$ to itself 4 times,” and where “multiply” means “abstract group multiply” which, for this group, means addition modulo 5). Indeed, we also see that we can write the seemingly strange equation $1^5=0,$ which just says that $1$ added to itself 5 times gives us the number $0$ (which is true because of how we cycle back through). We therefore see that every element in $\{0, 1, 2, 3, 4\}$ can be written as $1^m$ for some integer $m.$

How does this work in the group $\mathbb{Z}=\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ ? Well, we again see that the positive numbers can be obtained from $1$ by adding one to itself over and over again, so that for example $10=1^{10}.$ But how do we get the negative numbers? Well, I purposefully used the word “integer” at the end of the last paragraph, because if we let $m$ be negative in the expression $1^m$ then we indeed can obtain the negative numbers. Namely, in any group we let $g^{-1}$ denote the inverse of the element $g.$ Thus, in the group $\mathbb{Z}$ we see that $1^{-1}$ is negative 1, since the inverse of $1$ is $-1.$ Therefore, $1^{-10}$ simply means “multiply the inverse of 1 to itself 10 times.” However, “multiply” here means add, and so $1^{-10}$ is simply a shorthand notation for the instructions to “add $-1$ to itself 10 times,” and this gives precisely $-10$ ! Thus, any element in $\mathbb{Z}$ can be written as $1^m$ for some integer $m.$ Indeed, the integer $m$ that we must use is imply the integer itself! Namely, we have that $15=1^{15},$ $-27=1^{-27},$ and $548=1^{548}.$ We also need to make sure that we can get the element $0$ in the form $1^m,$ but this works precisely for $m=0$ since this would mean “add $1$ to itself zero times,” which precisely gives zero.

Thus, we see that the similarity between the integers, the groups of modular arithmetic, and the rotation groups is precisely that there is some element in each group that “generates” all the other elements in the group by simply multiplying itself over and over again. Indeed, there are many other such groups as we will see in the coming lessons, but we now have enough motivation to make the following general definition.

Definition 35.1 A group $G$ is called a cyclic group if there exists an element $g$ such that any element in the group can be expressed as $g^m$ for some integer $m.$ In this case, we call such an element (namely, the element $g$ ) a generator of the group $G.$

To recap, we have see that in the group of integers, this element $g$ is simply the number 1, in the group of rotations one such element is the element $\rho_1,$ and in the modular arithmetic groups one such element is the number $1$ again. I.e., the number 1 is a generator of $\mathbb{Z},$ the rotation $\rho_1$ is a generator of the group of rotations, and the number 1 is a generator of modular arithmetic groups. In the next lesson, we will explore how sometimes cyclic groups have more than one generator. For now, let’s leave this as an exercise: Does the group $\{e, \rho_1, \rho_2, \rho_3\}$ have a generator other than $\rho_1$ ? What about the group $\{0, 1, 2, 3, 4, 5\}$ ? What about the group $\mathbb{Z}$ of integers? If so, how many other generators are there?

Since we will answer these questions and give exercises in the next lesson, let us give this some thought and then move on without other exercises. (Yay! No homework!)

On To Lesson 36

Back to Lesson 34