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[00:00:00] Hello, my name is Shahriar Shahriari and this is a lecture in a series of lectures on undergraduate algebra based on my book Algebra in Action, a course in groups, rings, and fields. The purpose of this lecture is to talk about what a group is and to give you definition and examples. The plan for this lecture is to first tell you what a Cartesian product is, what's a binary operation, what's a group, and then go through some examples of groups. I had videos
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for each one of these, so I'm not going into much detail about them. The integers. The integers mod n. The group of units of integers mod n. The general linear group over a field. The symmetries of a regular n gon. Bijections from a set to itself. And continuous real functions on the real line. Except for the continuous functions, for each one of those, I have separate videos . Along the way, we'll also talk about what the order of a group is and what's an abelian group,
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and what a ring is, what a field is, and what the group of units of a ring are. These latter ones, if you don't follow every detail, it's okay, because when we do ring theory and field theory, [00:01:00] we will talk about these in much more general context, but here I will give you a taste of that as well. What's a direct product? The direct product of X and Y, or the Cartesian product of them,
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denoted by X cross Y, is another set. (x, y) is one element, the first entry of that pair comes from the set X, the second entry comes from Y. So two sets walk in. You create another set called X x Y. The direct product of X and Y is a new set consisting of ordered pairs of elements. The order matters. The first one is from X, and the second one is from Y. Here's an example. X is these three objects here, and Y is these two other objects. When I find X cross Y,
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I'm going to find pairs of elements, and so in the first entry I put something from X, and in the second entry I put something from Y, and these are all the things that I get, because there's three things in X, and two things in Y, I will get three times two, six elements. So X cross Y here is a [00:02:00] set with six elements, and these are it. Examples of direct products you've seen already. If R denotes the set of real numbers,
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then R cross R is what we call R^2. Any element of R2 is a pair of element s from R. Z/2Z is zero and one integers mod two. Z/3Z, its elements are 0 1 2 those are integers mod three, and then Z/2Z x Z/3Z, their direct product will have six elements. If you notice, the first entry is either zero or one because. Z mod 2Z has that, but the second entry could be 0, 1, or 2. What's a binary operation? I will start with an
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example of a binary operation and then I will tell you what it is in general. Addition and multiplication are binary operations for the integers. Blackboard Z is the set of integers. If I give you two integers, three and five, for example, you can add them, you can multiply them. You input a pair of elements and you get a new element. That's what a binary operation is, but the order matters also. For example,
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matrix multiplication is a binary operation on n by n [00:03:00] matrices. You bring two matrices in and you can multiply them, and get a new matrix. But in this case the order matters. It matters which one is the first matrix and which one is the second matrix. The formal definition of a binary operation is that you start with a set and a binary operation on that set is actually just a function. A function from the direct product of S and S to S. Elements of S x S are pairs of
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elements and you apply this function and that gives you one element of S. A binary operation is a rule that we apply to an ordered pair of elements of S, and you get a new element of S. If the name of the binary operation is star, so if the function is called star, you might want to find star of a, b, then in group theory, we write that as a star b. Or even forget the star and just write it as ab and think of it as multiplication, as a product.
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What's the definition of a group? If you have been watching my videos on examples of groups, you already know. This is not going to be a surprise, is just going to be a review, and if you haven't seen those examples, go look at those examples first.[00:04:00] You don't want to start this set of lectures with just the definition. You start with a non empty set and a binary operation on G. So you have a way of combining two elements and getting a third one. Then G together
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with this binary operation forms a group if we have certain properties. The first one is closure. If a and b are in G, then a star b is in G. This is actually part of the definition of a binary operation. And so I didn't really have to say it, but I want to say that for emphasis. You also have to have associativity. a star (b star c) should be the same as (a star b) star c. And this should be true no matter what elements of this set you pick.
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You should have something called identity. I like to call the identity e. When you walk in a group, you will know you're running into an identity, if it has the property that when you apply the operation, e star a, or a star e, you get a. Multiplying by e doesn't change anything and you get the other guy back. You have to have inverses. If you walk in the group and say, you, who's your inverse? Someone is not sitting there whose name is
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a [00:05:00] inverse. You've got to figure out which one that is. And it's something that when you multiply it by this fellow, either way, you get the identity. So an inverse of an element is something that when you multiply it by that element you get identity. You've gotta have closure, you've gotta have associativity, you've gotta have identity, and you have gotta have inverses to have a group.
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Sometimes we write this notation parentheses G comma star parentheses closed, and we'll say this guy is a group. That sentence, ( G, * ) is a group. What that means is that G together with this operation is a group. If there's no confusion about what the operation is, we'll just say the group G, just the name of the set, and you will know what the operation is. But sometimes there might be confusion. We might want to emphasize what exact operation
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are we talking about. In which case, instead of just saying G is a group, we'll say, parenthesis G, comma, whatever the operation is, parenthesis is a group. Often, instead of a star b, if the operation is star, we just write ab. Identity element, the group has to have an identity element, and I like to call it e. If there's too many groups [00:06:00] running around, then they have different identities, then I might say e sub G. Just as well, you could call it 1 or 1 sub G.
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If G is a group it's a finite group if it has a finite number of elements, and otherwise it's an infinite group. The order of G is the number of elements in G, and it's written as G with absolute value signs around it. If you have a set with absolute value signs around it, you don't read that as absolute value of G, you read that as the size of G, or the order of G. If G is infinite, of course that will be just infinite.
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Or we will think of it as the cardinality of that set if it's infinite. When we're talking about finite sets, order of G just means how many elements there are in it. In a group, ab and ba, a star b and b star a, are not necessarily equal, they don't have to be equal. If they are, then we say that a and b commute. Commute means they can move past each other. If all elements of the group commute, then the group is commutative. In group theory, we like to call them abelian.
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This is named after the Norwegian mathematician [00:07:00] Abel, Let me give you examples. I'll start with the integers, the ordinary integers. 0, plus or minus 1, plus or minus 2, and so on. Those are the integers, and, ordinary addition is an operation. You can take one element and another element and add them and you get something in the integers. Integers with plus is an abelian group. It's group, but it's more than that, it's abelian, because it's commutative.
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Identity is zero, and inverse of a is minus a. Inverse of 3 is minus 3. Why? Because if you add 3 and minus 3, you get the identity zero. What about multiplication? The integers with multiplication is not a group. You do have closure. You do have associativity. You have an identity. The identity is one. If you multiply one by anything nothing happens, you get that other element back. And you do have
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commutativity. But you do not have inverses. So for example, if you take 3 its inverse would be something that if you multiplied by it you would get one. You would like to say 1/3, but one third is not an integer. So among the integers, there's no inverse for three. The integers is an example of a ring, because it actually has two operations. It has plus and [00:08:00] multiplication. It's an example of a commutative ring with identity. When I say
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commutative ring, addition in a ring is always commutative, but the multiplication in a ring may or may not be commutative. And a commutative ring is one where the multiplication is commutative. And this has an identity, means that multiplication has an identity. So to be a ring, you have to have two operations. An example is the integers. Integers with plus is an abelian group. And integers with multiplication has closure and associativity. And we also have
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distributive laws. x(a+b) is xa + xb and so on. That's what a ring is, where with one operation you have an abelian group, with the other operation you have closure and associativity, and you have distributive laws. If the multiplication is also commutative, then it's called a commutative ring. If multiplication has an identity, then it's a commutative ring with identity. And the integers with plus and multiplication is an example of a
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commutative ring with identity, which in ring theory lectures we'll study in some detail. The set of real numbers with ordinary addition, that is [00:09:00] also an abelian group. Identity here is still zero, and inverse of a is minus a. Here, again, multiplication is not a group, but the only problem is zero, because zero is the only one that doesn't have an inverse among the real numbers. If you throw that out, we have this set R with a cross on top, that's the units of R.
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And that's an abelian group. That's an abelian group because you have an identity . The identity is 1. If you multiply 1 by any real number, you get it back. And any real number other than 0, but we throw away 0, has an inverse. So the real numbers is a little bit different than the integers in that all you had to do was throw out 0 to get a group with multiplication. This is an example of a field. In a field, just like a ring,
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a field is a particular case of a ring, it's a nicer kind of a ring, you have two operations, just like a ring, with addition you have an abelian group, a commutative group, but with multiplication you just have to throw away zero and then you also get an abelian group. And we also have distributive law, example of a field. Other examples of fields [00:10:00] are the set of rational numbers. What are the rational numbers? Those are real numbers that you can write as
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a ratio of two integers as long as the one at the bottom is not zero. And the complex numbers those are numbers that you can write as a+ bi, where a and b are real numbers, and i is the symbol with the property that i squared is minus one. You might say these are not actually numbers. They are. The definition of a number over time has expanded . Things have been included in the sisterhood of numbers as time has gone on,
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and people have realized that you can work with these just as numbers, they have the same kinds of properties, and they're very good at modeling different things. At some point we will talk about complex numbers in more de detail. These guys are both fields. The rational numbers and the complex numbers. What that means is that if you just look at addition as your operation for either rational numbers or complex numbers, you have abelian groups. And for both of those,
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the identity is zero. And the inverse of any number is it's negative. And if you throw away zero in either case, then you also get [00:11:00] abelian groups. But identity is one, because one times other things doesn't change, and the inverse of any number is one over a. And again, remember, we have thrown away zero, so we don't have the problem of dividing by zero. And we have distributive laws. We have been talking about rings as examples, but let me formalize that.
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What's a ring with identity? It means that R with, with addition is an abelian group. R with multiplication, all you need is closure and associativity. You don't need commutativity here. And you have a multiplicative identity, something that when you multiply by other things, nothing happens. And you have to have distributive law. If you have these, then you have a ring with identity. Elements of R with multiplication are not necessarily invertible.
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The invertible elements are called units. Those are elements that have inverses, that have some other element that when you multiply them, you get identity. And R with a x on top of it is going to denote the set of units, all those invertible elements. And if you have a ring with [00:12:00] identity, then the set of units is a group, called the group of units . of R. I have given examples in my videos on general linear group,
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as well as on integers modulo n. You're welcome to look at that, or you're welcome to try to prove this yourself. It's actually not that complicated to prove that is a group. A field is when you have two operations, just like a ring. So a field just like a ring has two operations, but in this case with addition, just like a ring, you have an abelian group, but with multiplication, you have a little bit more,
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actually, a lot more. When you throw away zero, then you get the group of units. And so the group of units is everything other than zero. And not only that it's a group, but it's an abelian group. A field is the nicest kind of a object with two operations. Examples that we will talk about are integers mod n, that's a ring, and n by n matrices with real entries is also a ring. I have a whole video, on integers mod n and another one on units of integers mod
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n, [00:13:00] and you should watch those. n is an integer. Z/nZ is just the number 0 1 2 all the way till n-1. So there's n elements. Z/nZ has n elements, and we have two operations. The operations are addition and multiplication, not the usual addition and multiplication, but addition and multiplication modulo n, what does that mean? It means that if you take two elements and you want to find a+b or a b, what you do is you find
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the usual a+b and a b, but then you find the remainder of that when divided by n. You take out as many n's as possible, and the remainder, which will be a number between 0 and n-1, that will be what a+b and a b mean in Z/nZ. Z/nZ with plus and multiplication with this addition and multiplication modulo n is a commutative ring with identity. Zero is the additive identity. One is the multiplicative identity. And because of that, Z/nZ with plus is an abelian group of
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order n. And I have a complete proof of that in a video, including a proof of associativity. You can write this [00:14:00] Z/nZ as a group generated by just the number 1. Generated by 1 means that you start with 1, and just keep applying the operation to that 1 and see what you get. So you start with 1, then 1 plus 1 is 2. 2 plus 1 is 3, 3 plus 1 is 4, and so on. But you have this relation that if you add n of these, you get back to 0. And when it's generated by one element,
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it's called a cyclic group. So Z/nZ is an example of a cyclic group. In a future video, we'll talk about cyclic groups in more detail. Because Z/nZ with plus and multiplication is a commutative ring with identity, we can look at its group of units, and, in my video on group of units, we prove that what are the units? Which are the elements of Z/nZ which under multiplication have inverses? And those are exactly those elements that are relatively prime to n,
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those elements whose greatest common divisor with n is one. And those form a group. This group of units of Z/nZ is an abelian group of order phi(n). Phi of n is the Euler phi function, Euler [00:15:00] totient function, just counts the number of elements between one and n-1 that are relatively prime to n. These are examples of groups, both Z/nZ with addition, and the group of units of Z/nZ are very good groups to get to know.
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Now let's talk about matrices. Start with a positive integer and a field. If this is your first time through algebra, whenever I say field, just think the rational numbers, the real numbers, the complex numbers, or Z mod pZ, where p is a prime, integers modulo p, when p is a prime number. And you'll be fine. Notation for n by n matrices over F, sometimes I use M_{n x n}(F), and sometimes I use F^{n times n}. So both of these are the set of n by n matrices with entries from F.
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And when the entries are from F, sometimes instead of saying entries from F, we say n by n matrices over F. Over F means the entries are from F. What are the operations here? We have matrix addition and matrix multiplication. The addition is [00:16:00] commutative, the multiplication is not. The set of n by n matrices over F with this addition and multiplication of matrices is a ring with identity. The additive identity, the thing that you add to things and nothing happens,
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is the n by n zero matrix, the matrix that has all zeros in it. And the multiplicative identity, the matrix that when you multiply by other things nothing happens, is the identity matrix, the n by n identity matrix. This is not a commutative ring. Addition, always in any ring, is commutative. But the multiplication here is not commutative. ab is not the same as ba, necessarily, when a and b are, are matrices.
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Because this guy is a ring, the set of matrices with plus is an abelian group. And we can also look at the units. Which matrices are invertible? Which matrices have an inverse? That set is denoted by GL(n, F), and I have a whole video on the general linear group, GL(n, F). GL(n, F) is the invertible n by n matrices over F. Sometimes instead of over F we just put slash F, but [00:17:00] we read that invertible n by n matrices over F,
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and that means n by n matrices that are invertible, but the entries are from this field F. This is a non abelian group. The group of units of any ring are a group in this case because matrix multiplication is not necessarily commutative. It's not commutative. We have a non abelian group. We have seen in the video on general linear group another group that's quite related and important. n is a positive integer and F is a field. GL(n, F), as we said,
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was invertible n by n matrices over F, and operational was matrix multiplication. SL(n,F) is those invertible n by n matrices over F whose determinant is equal to one and operation is still matrix multiplication. And this guy, SL(n,F), those invertible, n by n matrices, whose determinant is one, is a subgroup of GL(n, F). What does that mean? A subgroup of something means that you're a group, but you're sitting inside a bigger group with the same operation.
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It's just some of the elements of the bigger group that themselves form a group. When you multiply two. Invertible matrices whose determinant is [00:18:00] 1, you still get another matrix whose determinant is 1. So you have closure, you still have associativity, identity is in here, and inverses are in here. If you have a matrix whose determinant is 1, its inverse will also have determinant 1. This is another good example of a group.
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Symmetries of any geometric object form a group. So you can pick your favorite object and look at its symmetries. One example of that would be if you take a regular n-sided polygon in the plane. Regular means that you have an n-sided polygon where all the sides have the same size and all the angles are the same. D_{2n} is rigid symmetries of P_n. One example of that is when n is four. D_8 would be rigid symmetries of a square. A regular four-sided polygon would be a
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square . I have a whole video on D_8 that if you haven't watched, you definitely need to watch or at least know about D_8 before you proceed. This is called the dihedral group of order 2n. 2n here is the size of the group, is how many elements are in the group. Topologists actually call [00:19:00] the same group D_n. They focus on the fact that these are the symmetries of P_n, and they call it D_n. In group theory, we call it D_{2n}. So, What are the elements of this group?
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We have gone through this in some detail, actually a lot of detail, in the case n is 4 and the group D_8. That group D_8 is something that you really want to know very well, because whenever we want to talk about anything in group theory, the first thing you should think about is that, what does it say about D_8? What are the symmetries of a regular n sided polygon? Well, there are rotations. You can
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take your polygon and rotate it. You can go 2 pi over n. or twice that, or three times that or n-1 times that. If you do n times that, that's the same as doing zero. So there's n rotations for an n gon. For a square, for example, there are four rotations, zero, 90 degrees, 180 degrees, and 270 degrees. For an n gon, there's n rotations and there's n reflections. Depending on whether or not n is even the axis of symmetry for those reflections are gonna be
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different. If n is, even the axis are gonna be through [00:20:00] opposite vertices or through opposite sides. For example, for a square the axis of reflection are either diagonal ones through vertices or a vertical or a horizontal one. But if n is odd, then the axes of reflection are through a vertex and the midpoint of the opposite side. But there's n of them anyway. So there's n reflections and n rotations.
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What's the operation for symmetries? It's follow one symmetry by the other. If you rotate and then reflect, after you do that, you get another symmetry. And whatever that is, that's the product of those two symmetries. If the rotation by two pi over n radians, you call that rho, and delta is any of the reflections, you can write down what the elements of D_{2n} are. The rotations are the identity doing nothing, no rotation, zero degree rotation. Then rho, rho
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squared, rho^2 means rho rho, do two rotations, so you get four pi over n, and all the way to the rho^{n-1}, and then using that one reflection, you can find all the other ones by multiplying them by [00:21:00] these rotations. Again, go back, go to the video for D_8 for much more detail. The relations are that rho^n is e. rho was the rotations, and if you do it n times, you're going to get back to identity. So that's going to be the same as identity. And whatever reflection you
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picked, your delta, when you do it twice, you get back identity. So rho^n equals delta^2 equals e. There's another relation. that delta rho, that means do rotation first and then do a reflection, is the same as rho^{n-1} delta. These two things do not commute, but you can move the rho across delta, but at a price. You have to write rho^{n-1}. So this is another example of a group. The penultimate example are permutation groups, and symmetric group. So if you take any set,
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maybe an infinite set, and if you look at one to one onto functions from that set to itself, bijections on that set, they form a group, and the operation is going to be composition of functions. What does that mean? If f and g are permutations or one-to-one onto functions, then f [00:22:00] circle g of x is gonna be f(g(x)). With this function composition, this thing is a group. When you find permutations or one-to-one
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onto functions from a finite set to itself, that's an important case. [n] stands for the set with n elements, 1 through n and S_n is the set of one to one onto functions from this set with n elements to itself. It's an example of Perm(Omega), because here Omega is this set bracket n. Elements of S_n are permutations of these n things, and S_n is called the symmetric group of degree n. We have seen in the videos on symmetric group that the number of elements of this group is n
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factorial, and this group is not abelian, for n greater or equal to three. So S_3, for example, has six elements, S_4 has 24 elements, and so forth, and they're not abelian. For our ultimate example, I want to look at some functions. C^0(-infinity, infinity) is a notation for functions from R to R that are continuous. So the elements are continuous functions that have R both as their domain and [00:23:00] their co
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domain. Examples of that would be functions like y = e^x, y = x^3 - 2x, or y = sin^2(x). These are all functions that would be in C^0(-infinity, infinity). y = 1/x would not be because you can't plug in 0. Its domain is not all the real numbers. If you have two continuous functions with R as domain and codomain, you can add them. How do you do that? You can define f plus g as a function also from R to R. f+g is the name of this new function, and to tell you what it is,
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I have to tell you what it does to some real number. f+g, what does it do to x? f(x)+g(x).
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This is called pointwise addition of functions. You just add them. For example, how do you add e^x to sin(x)? Well, you get e^x+sin(x). That's function. It's nothing surprising. And this is an abelian group. It's closed. If you add two continuous functions, you get another continuous function. It's associative and it has an identity. The x axis, y = 0, that's a function, a continuous
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function. And if you add that to any of [00:24:00] these other functions, nothing happens. You can also define another operation. Take f and g, and f circle g is going to be function composition. f o g is again the name of another function. Which function? I'm going to tell you by telling you what it does to a real number. What's f circle g of x? It's f(g(x)). That's function composition. If we take these continuous functions from R
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to R and use these two operations, plus and composition, we get a ring with identity. Not a commutative ring with identity because function composition is not commutative. f(g(x)) is not the same as g(f(x)). If I have a water bottle and if I want to drink. If I first open it and then drink some water, it's very different than first drinking and then trying to open the water bottle. Composing two functions, doing one then the other, is not a commutative operation. But this is a ring
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with identity. And what's the identity? The function y = x. That whole function is the multiplicative identity. Because if you compose things with that function nothing happens. If you do some function and then apply this y = x, which [00:25:00] means just give me back whatever you had then nothing happens. We can look at the units . That would mean the invertible functions, but which functions have inverses? Those are continuous one to one onto functions
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from R to R. Those are a subgroup of the Perm(R). Perm(R), again, remember, was all one to one onto functions from R to R. The units of C^0(-infinity, infinity) are those permutations that are also continuous. That's also a group, and it will be a subgroup of Perm(R). I'm done with the examples. And from here on, we're going to develop the theory of groups. So all we're going to actually focus on is that definition. We have a non empty set,
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and one operation, and it's closed, it's associative, it has identity, and it has inverses, and we try to see what we can say about it. Now we have all these examples, which means that if we can say something profound, then that will be helpful because it'll be true about all of these examples. But to see if you're getting anywhere, to see if you're making any progress, you might want to think
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about these two guiding questions. One question is that, how do you prove that a group is abelian? Abelian [00:26:00] means it's commutative. A priori, without knowing any theory, you might think that, well, I've got to just try every multiplication of every two elements to see if the group is abelian or not. And we want to see if our theory will help with that. And in fact, it does. It will help us in deciding if a group is abelian.
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Our second guiding question is that for a particular n, how many different groups of order n are there. We have to think about what different means but how many groups of order five are there, how many groups of order six, how many groups of order eight, and so forth. As we go along, we will come back to these questions as a way of testing whether or not we really have done anything. Whether or not we have any more knowledge about abstract groups or not.
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Other videos in this series of lecture, the first one was a general one about what abstract algebra is. Then I went through a whole bunch of examples, one about symmetries of a square, about permutation groups and symmetric groups, about integers mod n and their units, about matrices, and the general linear group. This lecture was what is a group. After this, we'll talk about cancellation and some elementary properties of groups, and then cyclic groups and orders of elements,
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and [00:27:00] then we will continue from then on. This is the end of this lecture. Like my videos if you like them, subscribe to my channel if you want to be subjected to math videos on your feed and keep hydrated at all times.
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