 So, previous we've learned about a cyclic subgroup. Now we want to talk about the idea of a cyclic group itself. So let G be a group. We say that G is in fact a cyclic group. If there exists some element that's called little g, such that big g, the whole group is equal to the cyclic subgroup generated by G. So if that little element generates everything inside of the group G, we say that it is a cyclic group. And so of course, cyclic subgroups of themselves, cyclic groups in their own rights, but we say that a group is cyclic if it has a single generator. And that's what we call this element, this element that produced the whole group. We call it the generator of the group. Now let's look at an example of some cyclic groups for example here, right? Let's take the integers with respect to addition. We've seen previously that the set nz, which is the set of all multiples of n inside of the integers. So like for example, if you took 2z, we're talking about, you have all of 0, 2, 4, 6, 8, all of these even numbers. We also want their negatives as well. So you get negative 2 going downward. So 2z, we've proved previously that this is in fact a subgroup of z. But I want you to realize that this right here is just the cyclic subgroup of 2 generated inside of z. And so, and this is true in fact in general, that when you take the group in z, the subgroup, this is just the cyclic subgroup generated by n inside of the integers. And so again, 2z is an example of such a thing right here. Now in particular, we prove this for any integer n. Which there are some very important special cases to notice here. So for example, when you take n to be 0, you're just going to get 0z, which of course is the same thing as the cyclic subgroup generated by 0, which would just give you multiples of 0, which is just 0. This is the trivial group. This is the trivial subgroup. And I want you to be aware of this in general, that if you take the cyclic subgroup generated by the identity of the group, you'll always just end up with a trivial subgroup. But with the integers, what happens if you use as your, as your generator the element 1, right? What if you take 1z? 1z would be the cyclic subgroup generated by 1. But 1z, of course, it's just the same thing as, I mean, it's not what you put a little baby in. What I mean here by 1z is that this would be the entire integer group here. It's the entire, you get all multiples of 1 would be all integers. And so this shows us that the integers with respect to addition is a cyclic group. It's generated by 1. It's also generated by negative 1. It turns out that this has two generators. Both 1 and negative 1 will act like generators because the thing is when you have a cyclic group, generators are not necessarily unique. We'll get into that very, very soon here. But a group is going to have multiple generators. One clear point is that if you have a generator of a group, of a cyclic group, its inverse will also be a generator of that group. So both 1 and negative 1 act as generators for the integers. And so the set of integers under addition z is oftentimes referred to as the infinite cyclic group. And so if you ever hear me talk about the infinite cyclic group, I'm talking about the integers with respect to addition. Why does it get this other name? Well, we've actually talked about a lot of finite cyclic groups. We'll talk about some more. But when it comes to infinite cyclic groups, this group is unique. The integers under addition is the only infinite cyclic group, which is why we call it the infinite cyclic group. And what I mean is the only one, what I'm saying is, if you find a different infinite cyclic group, I can argue that it's essentially the same group as the integers under addition. In turn, we call isomorphism, something we'll define and talk about much more later in this series. So speaking of finite cyclic groups, let's give a conversation about those. Consider the group zn under addition. We can, I don't actually remember if we talked about these or not, but we can define them similarly. If you talk about the group m zn, this is gonna be the multiples of m inside of zn. So there's gonna be reduction mod n in that consideration. But if you take m zn, you just take the multiples of m in the group zn, this is just the cyclic group generated by m. And so this, we'll give you an example of such a thing. Let's consider the group z6. And if you take two z6, we're looking for those multiples of two when you're working mod six. So you get zero, two, and four. Notice that the next one would be six, which is just zero again. The next one would be eight, which is just two again. The next one would be 10, which is just four again. And just repeat itself over and over and over again. So two z6 is this subgroup right here. And this right here is just, that's a really long equal sign in case you're wondering, this is just the subgroup generated by two viewed as a subgroup of z6. So this is the subgroup generated by two. But also this is the subgroup generated by four as two and four inverses of each other. Two and four will both generate that. So let's think about that one for a second. If you're looking at zero, the subgroup generated four, you're gonna have zero, you're gonna have four. You're gonna have four plus four, which is eight, but reduces mod six to be two. You're then gonna have two plus four, which is six, and you get the identity and it'll cycle around each other over and over and over again. This is actually why we call them cyclic groups or people in the UK sometimes call it cyclic groups and things like that, sort of a European pronunciation. But the idea is the elements of this group form a cycle. If we think of it kind of like the following, right? You take two plus, zero plus two is gonna give you two. Then you take two plus two, which gives you four. And then you're gonna get four plus two, which gives you zero. Then you're gonna take zero plus two, which is two again and then to four and then all the cycle, as you add, as you add, as you add two, two, two, two, two, two over and over and over again, you just spin around in the cycle over and over and over and over and over and over again. And, whoops, put that line back there. And then taking negative two, because you had the inverses just means you're going in the cycle backwards. So are you going like a clockwise or counterclockwise rotation? That's all that's going on there. And so the same thing happens with four. When you add four to things, you're just moving on this same cycle over backwards, backwards, backwards, backwards. And so cyclic groups get their name because of this sort of cyclic pattern, this pattern of cycles that's happening right here. Now, this is just an example of subgroups when we look at Z6. In general though, just like with the integers, if we take the subgroup generated by one inside of Zn, that'll produce every single element in the group. And so the group Zn is likewise generated by one, just like the infinite cyclic group, the integer Z. That also means it'll be generated by negative one, which really is just gonna be N minus one when you work mod N right here. Now, it turns out though, on the other hand, unlike the integers whose only generators were one and two, when you work with integers mod N, you can actually get lots of different generators. It turns out that they can be quite common. So for example, we work with Z5. I claim, you know, so Z5, this is the subgroup generated by one, which would look like zero, one, one plus one, which is two, plus one, which is three, plus one, which is four, plus one, which is five, up that's just zero, you get everything. But this is also the same thing as two, the cyclic subgroup generated by two. The idea here is the following, you get zero, you get two, plus two, which is four, plus two, which is one, plus two, which is three, plus two, which is zero. Notice how two got everything as well. But not just two, what if we look at the cyclic subgroup generated by three? You're gonna get zero, three, six, which of course is one, plus three, which is four, plus three, which would give you seven, which reduces to two, and then two plus three is five, right? What about four? The cyclic subgroup generated by four. Turns out that's gonna do it as well. You get zero, you're gonna get four, plus four is eight, which is three, plus four, which is seven, which is the same thing as two, plus four, which gives you six, which reduces to one, and then four plus five, sorry, four plus one is then five, you see like this. So when it comes to the cyclic group of order five, aka Z five, seems like every number was a generator. There was of course one notable exception. If you take the subgroup generated by zero, this will only produce the trivial subgroup. So zero, the identity can't be the generator here, but you get that every other number, one, two, three, four are generators. And then one other example to kind of compare this to, if we look at Z six, you get the subgroup generated by one, this is gonna give you zero, one, two, three, four, five, and then six again, right? And then you take the subgroup generated by negative one, which of course is just five in this context. You're gonna get zero, five, then you'd get 10, which reduces to four, then you're going to get nine, which reduces to three, then you're going to get eight, which reduces to two, then you add five to that, you get seven, which reduces to one, and then five plus one is six. So those are both generators, but on the other hand, if you look at other elements, like say the cyclic subgroup generated by two, we already saw that that was the same thing as zero, two, and four. I want you to convince yourself that would be the same thing as the subgroup generated by four, in terms of mod six, right? We said that one. If you take the subgroup generated by three, that's the only one we haven't considered yet, other than zero itself. In that case, you're gonna get zero and three, three plus three is six. So two, four, and three don't, they don't produce everything. They only produce, they produce proper subgroups, but one and five produce the entire thing. So determining whether a number is a generator for a finite cyclic group is a little bit different thing. And it turns out it has a lot to do with GCDs, whether a number is going to be a generator or not. So for example, the generators of Zn are gonna be all those elements which are co-prime to n. Notice what we have here, that one and five are co-prime to six. They generated two, three, and four share a common divisor of two, three, and two, respectively. So then of course zero also has a common divisor with six, which would be the GCD would be six. So one and five were co-prime, two, four, and three were not co-prime. But when you look at like five, for example, five's a prime number, so other than zero all the numbers were co-prime to five. So one, two, three, and four. And so what we see happening, what we see happening in general then is that if you have a number P, which is co-prime to n, then by the Euclidean algorithm, you can write one, which is the GCD as a linear combination of P and n. So you get some AP plus BN. But if you reduce this number mod n, you're going to get AP. So this tells you that one is a multiple of P. So if we kind of summarize the observation of the Euclidean algorithm here, if you take the cyclic subgroup generated by P, this of course will contain the element AP, which we're saying in this cyclic group is the same thing as one. And so since the cyclic subgroup generated by P contains one, it has to contain the cyclic subgroup generated by one. But as we mentioned earlier, the cyclic subgroup generated by one is all of ZN. And as the cyclic subgroup is contained in ZN itself, equality is forced. And you see that ZN is generated by this element P. So anything that's co-prime to the modulus for the cyclic subgroup will be a generator. Anything that has a common divisor will not be a generator. And that's the basic argument for finding generators for finite cyclic groups.