 Welcome back to our lecture series Math 42-20, Abstract Algebra 1 for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. In this lecture, lecture 7, we start chapter 3, which in this chapter we're going to introduce the most fundamental and most important structure in Abstract Algebra called the group. And I should mention that in algebra and mathematics in general, but especially in algebra, we're not always the most clever in terms of the things that we name. So I mean, after all, what's a set, right? A set, if you think of just like English, it's just a collection of things. Okay, that makes sense when you think of it from a mathematical setting. But what's a group? Well, a group is a collection of things. We're going to see other things like that as well. We're going to talk about a field, which that might make you think of like a meadow, right? But if you talk about like, oh, all of the experts in my field, a field can also be used as a synonym for group. It's a collection of things. Also, we'll talk about rings. What's a ring? Well, again, you might think about some shiny thing you want on your finger. But if you think of like a ring of thieves, again, the word ring and some connotations in some context can mean a collection of things. So you're going to see that a lot, a lot when one talks about groups and things. But before we define what a group is, what I want to first introduce is the idea of a binary operation. Algebra is all about studying operations, binary operations being one of the ones we talk about the most. What is a binary operation? So if we have some set, which we're going to call a G, thinking of this as a prototype of what a group is going to be, a binary operation is a set, or is a function, excuse me, from G cross G to G. So basically, you take two elements of your set G and then you produce a third element, a binary operation. We're quite accustomed to things like this and we'll define what that means in just a second. Let's say that the binary operation has the name circle. So there's going to be some symbol to denote the function. Oftentimes, when we describe the function, you're going to have here some element of the domain a, b, it's just two elements from the set G. The image of a, b under this map circle, we often denote it as the following, a circle, b. That is, we put the symbol for the operation between the two elements we're acting on. This is similar to how one denotes relations. This is how we denote this. Now, there are also situations where we put no symbol between them whatsoever. This is very common when we do multiplication. So juxtaposition can be used to describe the operation in hand here. So the map circle will map a, b to the element a, circle, b. Yeah. So we see this have a notation all the time. And so we'll often write as a pair G, common operation. This represents that it is a set with the binary operation of circle. And so in the, in the algebra community, this is often referred to as a magma. That's not a term you see a whole lot because we don't often focus on that, but it's just a set with a binary operation. And we're actually quite accustomed to many examples of things like this. Take, for example, the sets, the natural numbers, the integers, the rationals, the reels, the complex numbers, for example, these are all sets with the binary operation of addition. So addition is an example of a binary operation. If you add together two natural numbers, you get back a natural number. If you add two integers, you get an integer. If you get, if you add two rationals, two reels, two complex numbers, you always get back an integer. Not an integer, you always get back a number of that type. And I should also mention that multiplication is also an example of a binary operation on this set here. If you take any two natural numbers and multiply them together, you get a natural number. Same thing for integers, same thing for rational, same thing for reels, same thing for complex numbers. This idea of addition and multiplication always gives us binary operations. This is also true for the set ZN, right? So ZN is the set of integers, it's a set of congruence classes of integers mod N. One can denote the notions of modular addition and modular multiplication. Now sometimes people draw a circle plus or circle times to denote this. Just in terms of latex, you can actually write this backslash O plus and O times if you want to do that symbol there. I should also mention that for the standard, if you want to do this like X symbol in latex, that itself is just going to be backslash times. I think that's the command off the top of my head. I hope that's not wrong, but backslash times to do it. I myself am not, don't necessarily feel like I need a symbol to distinguish modular addition from regular addition because by context it'll be clear what we're referring to. And that's mostly because I'm an algebraist and therefore that's kind of how I feel about some of these things. So these are all important sets for which addition and multiplication make great binary operations. Some other sets we should mention here, if like if you take vector spaces for example, you take the vector space RN. This would be the set of all column vectors of the form, you have like an X1, X2 up to XN, where all of the XIs, these are real numbers. So you take this vector space in terms of vector addition, right? This is a binary operation, vector addition. Because in vector addition what you're doing is you're going to take two vectors from RN and you combine them together to produce a vector in RN. And it's just the usual rule that you add components. The first components get added together, the second components get added together, the third components get added together, etc., etc. Now as a non-example, we can mention something like scalar multiplication. If you take scalar multiplication of vectors, I should mention that this is not a binary operation. And why is that? Scalar multiplication, what you do is you're going to take a scalar R and you're going to combine that with some vector RN to produce a vector in RN. And so notice here, we don't call this a binary operation because one of the operands is not actually an element of the set. R and RN are going to be different sets. These are not the same type of object. This isn't like an integer plus an integer. This would be like an integer plus a matrix, right? Or something like that. This takes two different objects and combines it to create a vector, right? This is not an example of a binary operation. So this is not a binary operation. These type of objects are of importance to, of course, algebra. That's why we care about scalar multiplication here. It's not a binary operation. But you know, this is something, we'll talk about this maybe, well probably in actually in 4230, I think. This is a type of thing we could call an action. It's a ring action of some kind. But again, that's not something we need to get into right now. Focus on binary operations. Scalar multiplication is not a binary operation. And so other operations we do see in linear algebra probably don't fall under binary operations. If you think of the dot product, for example, the dot product, same type of idea, the dot product, as it's usually defined, it's going to be a map from RN times RN. But then it produces something in R when you're done, right? So you take two things that belong to the vector space, combine them, but then you get a scalar when you're done. So again, this is another example, a non-example of a binary operation. This will be something you call like a bilinear form in linear algebra, but it's not a binary operation. Binary operation requires that all three sets in consideration are one and the same thing. You take two elements from a set, combine them together to produce a third element of that same set. Things like scalar multiplication, dot product kind of leave that setting. On the other hand, if you take, for example, the set, if you look at R3, R3, you actually could use the cross product. That's something that you often see in linear algebra. The cross product is an example of a binary operation. Because you take a vector in R3, you combine it with another vector in R3 and that produces a vector in R3. So that gives us, that does give us a binary operation. That's kind of special to R3. All right. Some other operations I want to mention. Let's take like matrix multiplication without leaving without leaving linear algebra first yet. If you take the set M sub N of R, sometimes people denote this as like R to the N cross N. This right here is going to be the set, the set of N by N matrices, matrices with real entries. You can have a binary operation with respect to addition, right? So matrix addition, that's of course going to be an operation, a binary operation. Now that, that's going to be true for any sort of N by M matrix. Nothing particularly special about that. I mostly want to focus on matrix multiplication for a moment, right? We could talk about the product of any two matrices, right? So if you take a matrix A times B, how that gives you another matrix. And so matrix multiplication, get M times N times R cross M N R. This will then produce an M by N matrix. So if you take square matrices and multiply them together, that gives you an example of a binary operation. Now one has to be sort of careful because matrix multiplication is defined for other shapes of matrices that you can take, you can take for example, a three by two matrix times a two by three matrix. And that does produce, for example, a three by three matrix. So there are ways of defining matrix products in some situations. But the problem for the problem about matrix multiplication in general is if you take like a three by two matrix, you want times that by four by seven matrix, that's a no go. There's no such matrix like that. So the issue kind of there with matrix multiplication in general is that not every possible product is defined on the set, right? If you look at, if you want to like the set of all matrices times all matrices, you want to produce a matrix without any restriction on the dimensions of the matrix. Well, the problem is there's going to be some ordered pairs for which it's not defined. So it's not even a function at that stage. And so these are some, these are some important counter examples you should mention about binary operations. So like we saw with scaling multiplication, it could be that one of the operands is from a different set. So that doesn't make it an operation. It could be that the resultant is actually not part of the set. And therefore it's not a binary operation. Or another common thing that one talks about is that it could be that, yes, you have some type of way of combining two elements together from the same set to get a third element, but not every possible combination is defined. This kind of leads to, and we talked about like with bilaternary forms, scalar, multiplications, a group action. This kind of leads to the idea of a group void, which these are all like generalizations of the notion of a group, which we're going to try to talk about. And you don't have to worry about all these generalizations in this course, but there is relevant algebraic stuff going on there. But I should mention that as defined, these things are not binary operations. And one other example I do want to point out, matrix multiplication actually is just a special case of function composition, because matrices can be associated to linear transformations for which the matrix multiplication is just the composition of the two functions. And so when you have something like recall that the set B to the A, this is the set of all functions of the form F, all functions that go from A to B, like so, we can define an operation, right? We can compose functions together. You know, we'll have things like maybe, you know, F as a function from A to B, and G is a function from B to C. Then it makes sense to compose the two functions, G of F, which will be an element of C to the A. We can compose these things together. It kind of gets to the same problem that matrix multiplication does. Not every possible composite of two functions is well defined. But in the special case, like in the in my case, if we take this case of X to the X, right, so this would be the set of all functions F of the form X to X. These are functions which map back the same set back into itself. In this situation, we now do have a binary operation. In this setting, right, in this setting composition, basically only in this setting, of course, composition in this situation is in fact a binary operation. So you have to make sure that binary operation, that the two operands and the resultants all belong to the same set, and every possible pairing of operands gives you a well defined resultant. That's what we need to be a binary operation.