 We are now in a setting which we can define what we mean in abstract algebra by a group. So a group is a set equipped with a binary operation that satisfies the following three properties which we call axioms. From a logical point of view, the axioms are gonna use fundamental truths that must be true in order for the object to be a group. So it's not just good enough to have a set with a binary operation that binary operation has to satisfy these three conditions to be a group. And those three axioms are the following. The first one is called the associativity axiom. And as you're probably familiar with associative operations, the associativity axiom tells us that if we take any three elements inside the set, we'll call them G, H, and K, then it must be true that if we operate, because these are binary operations at all, they're not trinary operations. In order to operate, we have to take two elements at a time. So the associativity axiom tells us what we can do when we have three elements. So the associativity tells us that if we do the second two first and then follow it up with the first element, that's the same thing as doing the first two elements first and then following up with the third element right there. Or oftentimes we say that we can re-associate the parentheses, we can put them in in either order. And because of the associativity axiom, we can actually turn our binary operation into a trinary operation. We can say things like G composed of H composed with K here, whatever the operation is. And we can omit the parentheses without ambiguity because the two operations give us the same thing. A group must be associative. Now I want to caution you that associativity is not an automatic operation. Like if we think of just integers for a moment, right? Integers have an operation of subtraction. That is a binary operation, but subtraction is not an associative operation. If we take something like six minus three minus one, you'll notice that this by order of operations you get six minus two, which is equal to four. On the other hand, six minus three minus one, this right here is equal to three minus one, which is equal to two. They don't match up because addition is a non-associative operation. This is true for division. This is true for like the cross product in R3. Not every operation is associative, but to be in a group, we do require associative, whoops, I didn't want all that. Erase, erase, erase, erase. We do require the operation to be associative. The next one, the identity axiom. The identity axiom says that there exists an element in the group. This is a distinguished element. There's an element in the group, which oftentimes we call it E. This actually comes from German where the word for identity there starts with an E. We take an element E so that if you take any other element in the group that G operated on this identity element E is the same thing as E operated on G, which is just G itself. Then when you operate by the element, this so-called identity element, nothing happens to the element, the other element in play here. So this is very true for like addition, right? Typically if you take like X plus zero, this is equal to X, where zero is the additive identity, or if you take one times X in terms of multiplication, or say X is just a real number here, one times X is X, one is this multiplicative identity. In terms of matrix operations, if you take A times the identity matrix, this is just equal to A, where here the identity matrix is the identity in terms of matrix multiplication. If you have a function composed with the identity function, right? Where the identity function just maps X to X. This has the property that F composed with the identity is just F, it's the identity of that operation. So in a group, we require that the operation in play has an identity. Now, not every operation has an identity, right? Take subtraction as another example, right? If let's take an element like seven, right? Is there an element such that E minus seven equals seven? Now you might be like, oh, that's easy, just add seven of both sides and E would have to equal 14. But the identity element has to be universal. That is every element in G has to have the same identity element. So yeah, you found something that works for seven, but let's switch it up to eight, right? E minus eight, that would be 14 minus eight this time. And you're gonna see that gives you a six this time, which is not eight. So in fact, there is no universal element that's an identity for subtraction. Well, I should say that there is no left identity for subtraction. It does work much easier if you work from the right. What do you mean by that? If you take X minus zero, that's always equal to zero. So subtraction has a right identity, but it doesn't have a left identity. There's an element which works in the right operand that acts like an identity, but there's not an element that works in the left operand. So some funky things can happen in terms of identities. There could be no identity. There could be a left identity, but no right identity. There could be a right identity or no left identity. Or of course, there's this two-sided identity, which we call an identity. The group requires we have this two-sided identity. That's why we listed both properties as a right identity and a left identity. Not every operation has those things. And the third and final axiom of group is the inverse axiom, which is sort of a strengthening of the identity axiom that for any element in the group, there is a distinguished element called G inverse, which we call the inverse of G. And it has the property that when you operate on G in its inverse, the inverse could come second, the inverse could come first, it doesn't matter, but in either situation, when you operate by the inverse, you end up with the identity of the group again. So that's what we mean by an inverse. When you combine the two elements together via the binary operation, you get back the identity. A group has to have inverses for each element. Now, do pay attention to the quantifiers here, right? So with the quantifiers, this says that there exists an element. So this element E is independent of the group, independent of other elements, right? Because there exist an element such that for all elements, this thing happens. With the inverse statement, the quantifiers are reversed. For all elements G, there exists. This inverse element is not universal, right? The inverse element depends on the element in question. What's universal about it is that if you give me any element of the group, it'll have an inverse, but different elements will likely have different inverses. And for a general binary operation, inverses are not guaranteed. Take subtraction again as an example, right? We saw that there is an identity zero, although it's only a right identity. Can we come up with inverses of this? Well, again, if you take like X equals seven, you could always take seven minus seven, which equals zero. And so that tells us that the inverse of seven is itself. And that's gonna happen in general. If you take eight minus eight, right? That's always gonna equal zero. If you think of this value fixed, but it doesn't exactly work the other way around. Well, it's different, right? If you have the equation like X minus seven equals the identity zero, well, then you're gonna have to plug in seven right here. And the inverse will always just be the number itself, right? You know, if you switch that to an eight, you'll just plug in an eight right here. That's how you do it. So you have a left inverse. Like the left inverse is always just equal to the number itself. But if I switch things up, what if we now put seven minus X right over here? You wanna solve this equation. In this situation, right? You would still plug in the thing there. So you get like this, you get this like left and right inverse. That's how it works. But again, it's only a one-sided inverse. There's things like that. Another operation we could think of for a moment. Let's play the following game where S equals the numbers one, two, three, four, five. In this situation, let's take as our operation the max operation. So that means is like the max, you're gonna take the biggest of the numbers, right? So in this situation, if you take like say one max, we're gonna write like this, one max any other number. Notice you're always gonna get back that number because one is the smallest number. So in this situation, one is an identity element, all right? One's the identity. On the other hand, if we take something like, if we take something like say three, what can operate on three to give me a one? Well, in this situation, this binary operation can't do that, right? Because if I stick in one, that's gonna give me back a three, right? If I stick in a two, that's gonna give me back a three. If I stick in a three, that's gonna give me back a three. If you stick in a four, you'll get four. And if you stick in a five, you're gonna get back a five. No operation on three will ever give you a one. So this is an example of a set whose binary operation has an identity, but it doesn't have an inverse, right? So these conditions are independent of each other. That is having one doesn't necessarily guarantee the other. The sociativity, the identities, and the inverses work together to form what we call group theory. And to be a group, your binary operation on the set has to satisfy these three conditions. Now, that doesn't mean there aren't other conditions that the set, the operation could have. If additionally, there's a fourth condition, the commutative principle, right? Where if you take any two elements, G with H, that the order in which you operate doesn't matter. G composed with, you know, G times H is equal to H times G. It doesn't matter the order. We call this a commutative operation. And if you have a group that's commutative, this is often called an abelian group. Now, an abelian groups, well, I should say that in groups in general, we often use multiplication notation for groups, in which case we'll just call it like times, just in the usual sense. With abelian groups, we often denote the operation generically with a plus sign, in which case the identity element would be a zero. If you're talking about a multiplicative group, your identity element would typically be called a one. Or again, you can call it E if you want to be even more abstract. The inverses of an element are going to typically be drawn X to the inverse, right? But an abelian group, you often write negative A because you think of it as the negative number associated to that. This will be more clear in examples we talk about. But some examples of groups that I want to present to you in this video here. The group, so we have the integers with respect to addition, that's a group. And the rational numbers with respect to addition, that's a group. The real numbers with respect to addition is a group. And the complex numbers with respect to addition is a group. You might be like, oh, everything with addition is a group. That's not exactly true. If you take the natural numbers with respect to addition, that's not a group. The reason is, addition in all cases, in all of these cases is associative. And in all of these cases, there is an identity element, zero. But the problem with the natural numbers is there's no inverses. If you have something like, say, three, three doesn't have an additive inverse. Is there an element that we can add, key word there, is there an element we can add to three to get zero, right? In the integers, you'd plug in a negative three there, but the natural numbers don't have such an element. Inverses do not exist in the natural numbers, and therefore we don't have a group operation there. Let's see, I should also mention another sort of non-example here. If we take the set of positive integers with respect to addition, this also doesn't form a group. Basically the same reason, right? You don't have inverses if you don't have positive integers. You also don't have an identity. And so restricting any of these sets only to be like positive rationals, right? That wouldn't work, positive integers or positive reals that wouldn't work. Positive complex numbers, what does that even mean? But restricting can be, you have to be a little bit careful because you might not get, you might not get a group if you restrict the sets. So those are some important examples. In terms of multiplication, right? If you take Q star, we talked about this one before with respect to multiplication. That is a group, the non-zero reals with respect to multiplication forms a group. The non-zero complex numbers with respect to multiplication forms a group. In all these situations, multiplication of rationals, reals and complexes is associative. There's an identity element, the one, of course. And then there are inverses, and particularly just take the reciprocal of each of these things. Some important non-examples, of course, if you take like the natural numbers with respect to multiplication, doesn't work. Because although multiplication there is associative, you have an identity, you don't have inverses. But that's also true for integers in general, right? It has nothing to do with positive or negative anymore. If you take the integers, likewise, you don't always get inverses for integers. Oftentimes, you don't have inverses. And therefore that doesn't form a group there as well. And same thing, restricting to positive quantities doesn't exactly help you here. If you take positive integers, you're still gonna have a problem. Doesn't work. But I should mention, on the other hand, if you take, for example, positive rational numbers with respect to addition, multiplication, I mean, that is in fact a group. The product of two rational numbers is in fact, is positive rational. And that's also true for reals and complex numbers. This is sort of a group sitting inside of a group. Something we'll talk about a little bit later. Some other non-examples we should mention, like the integers with respect to subtraction that doesn't form a group, like we mentioned. Subtractions a non-associative operation and therefore weird things can happen. Associativity is a very, very powerful axiom to have here. The rational numbers with respect to subtraction doesn't work. The reals with respect to subtraction doesn't work. Let's see the complex numbers with respect to subtraction. None of these are gonna be groups because subtraction is a non-associative operation. Same thing with division, right? If you try to do something like q star with respect to addition, sorry, division, that doesn't work, r star with division, c star with respect to division, these are not groups because your operation is non-associative. It has to be associative. An important example, if we take the set of functions from x to x and we take function composition, this is an example of a group. Another example, if you take s sub x, this would be the set of permutations from x to x. This is also an example of a group. And again, this is a group sitting inside of another group. This second group is commonly referred to as the symmetric group, which is actually what we call s sub x, the symmetric group. And this will be a very important group that we will study this semester, I should say in this course. Now I should mention that all of these groups we said previously, right? All of these green groups we said before, these, the like addition with integers, rationals, reels, complex, multiplication of rationals, reels, and complex, these are all examples of abelian groups because addition and multiplication are commutative operations. On the other hand, when it comes to this function composition and the symmetric group, these are examples where the operation is non-commutative. So we refer to this as a non-abelian group. These are both non-abelian groups. The operation does not have to be commutative. In this, I wanna present one more example. I tend to technically, important groups we're gonna see. This one's a little bit more foreign. So I didn't wanna actually have it typed up nicely for this example here. Let's take the set Zn, that is the set Zn. This is gonna be the set of congruence classes of integers mod n. So you see things like the class containing zero, the class containing one, the class containing two, the class containing up to, you know, n minus one. Now I should mention that oftentimes it gets tedious to write these classes with the brackets. Sometimes people write bars, one bar, two bar, three bar, what have you. So oftentimes when someone describes Zn, you'll notice they'll just write the integers zero through n minus one, knowing that each number is just a representative of the congruence class. This is a very convenient notation, but one has to be careful because if we start using representatives to represent the class, we have to make sure that we've defined operations when we define functions on this set that they're well-defined, that the formula is not dependent on the representative, but instead it gives us the class itself. Now with respect to modular addition, Zn does form a group. Modular addition is associative, basically for the same reason that integer addition is associative. It's also a commutative operation. Again, this is a consequence of the associativity and commutivity of the integer addition with the division algorithm. There is an identity element for modular addition. It's just the congruence class contains zero, that is multiples of n. There is, of course, also an inverse class. The class that contains the additive inverse with respect to regular addition is also the inverse element with respect to modular addition. And so Zn forms a group. And so we would write this as like Zn comma plus, or if you wanna do circle plus to emphasize that, that's also great. Well, so this is an example of an Abelian group with respect to addition. Now, like we saw with like q, r, and c, we can make a group with respect to multiplication, but we have to throw out zero, right? In terms of Zn, we have to throw a little bit more out than that. To make a Zn a group with respect to multiplication, we're gonna define a set which we call Zn star. And Zn star, we're gonna grab all of the integers which are co-prime to n. So we want those integers in Zn, which are relatively prime to one, the relative prime to n, the GCD of n and k is equal to one. Now, the notation we're gonna use in our lecture series is Zn star to kind of mimic the notation we had earlier. In some textbooks, such as Judson's, they use the notation un to denote the same set. And the idea is if you think of Zn as a ring, which we haven't defined with that mean yet, this would be the group of units of that ring. So it's the U sub n right there. We're gonna do Zn star. And so these are gonna be the invertible integers mod n. That is, these are the integers which have reciprocals. And so Zn star with respect to multiplication, not addition, but with respect to multiplication, this will form an abelian group. So the fact that multiplication modularly is associative and commutative will be, again, from the fact that integer multiplication is associative and commutative. And we can carry this through with the division algorithm. We will have an identity element. This is gonna be the congruence class that contains one. Now inverses can be a little bit of a trick. How does one prove that every element has an inverse? Turns out the answer there is gonna be to use the Euclidean algorithm. Because the Euclidean algorithm tells us that if two numbers are co-prime, so if k and n are co-prime, there's some linear combination that will add up to be one. So ak plus bn is equal to one for some b and some k. Well, if you move the bn to the other side, right, you see that ak is equal to one minus bn. Now bn is a multiple of n, right? It's kind of in the name there. And so if you reduce that mod n, you're gonna see that ak is congruent to one mod n. One is the multiplicative identity mod n. Therefore, k here, k would then give you the multiplicative inverse of a mod n. And so this will, of course, depend on the modulus in itself. And we'll do some examples of this in the next video. But z and star is an important, it's an important abelian group as well. Z and star, this is an abelian group and it has, it's associative, it's commutative, it has identity, and it also has inverses, although the inverses take the Euclidean algorithm to find.