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All right, so hello everyone. I'm Norman Wildberger, and we're at the University of New South Wales today. This course is on the history of mathematics, and I want to talk about group theory, which is a very interesting development. It started in the 19th century and then grew and grew, becoming ever larger and more important, eventually becoming a very significant contributor to 20th-century mathematics and also physics.
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So, roughly, we can say that group theory has its origins in algebra
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and perhaps in the theory of equations. Its most famous champion is Évariste Galois, a brilliant young French mathematician who was really the first to grasp the idea of a group. This is an introductory talk, and I'm not going to require that you already know what a group is. I'm going to explain to you what a group is if you don't already know or if you've forgotten.
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This is really the first of many objects in abstract algebra. It led to the development of what we now call modern or abstract algebra, which ended up becoming a major theme in 20th-century mathematics. It involves a lot more than just groups; it also includes other abstract algebraic objects such as fields, rings, algebras, vector spaces, Lie groups, Lie algebras,
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and quite a few other more exotic kinds of abstract structures. The pattern, I suppose, is similar in many of these topics. We start off with practical examples that come up in some kind of situation, then abstract some minimal structure that captures the specific situation, and finally, we end up studying the abstract objects or abstract theory sometimes almost independent of the original examples. Initially, we are interested in going
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back to the initial motivating examples, but ultimately, once the theory gets big enough, it becomes interesting in its own right. Group theory is an excellent example of all this.
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The story really starts back in the 18th century, probably with Euler, as many things do. In 1758, Euler was thinking about a result of Fermat—Fermat's Little Theorem—which I remind you states that if you have a prime number "p" and some number "a" which is
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relatively prime to "p", then "a" raised to the power of "p minus 1" is congruent to 1 modulo "p". So, that's what this means. This means "a" and "p" are relatively prime. Then we have this theorem of Fermat, a very famous and important theorem that states that if we take "a" and raise it to the power of "p minus 1", then it is congruent to 1 modulo "p". Now, Euler was interested in proving
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this. He was also interested, alternatively, in generalizing it. So, Euler found more generally that if we replace "p" with another number, say "n", for an arbitrary "n" (a natural number), we need to define Phi(n) to be the number of elements in the set "k" (belonging to 1
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to n-1) such that "k" and "n" are relatively prime. This is called the Euler Phi function. This measures how many numbers less than "n" are actually relatively prime to "n". So, for example, Phi of five is four because there are four numbers (all the numbers less than five) that are relatively prime to five. Also, Phi of eight is four because if we look at the numbers between one and seven, then one, three, five, and seven are exactly those that are relatively prime
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to eight. This is what's called the Euler Phi function. Euler then came up with this beautiful extension of Fermat's result that says if "a" and "n" are relatively prime, then "a" to the power of Phi(n) is congruent to 1 modulo "n". It was really along the way of investigating these things that Euler came up with some algebraic structures that ended up being simple examples of groups. So, let's have a look at what those structures are. There are some numbers
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less than "n" and relatively prime to "n" that have an algebraic structure.
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They are closed under multiplication modulo "n". Let's have a look at what we're talking about. Let's first consider the case when "n" is the prime number five. In that case, there are four numbers less than five which are relatively prime to five. This is the multiplication table for all of these four numbers, where we're going to do multiplication mod five. I remind you that means we ignore all multiples of five or we reduce things mod five by
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throwing away multiples of five. Otherwise, it's just like ordinary multiplication. So, 1 times 1 is 1, 1 times 2 is 2, and so on. 2 times 1 is 2, 3 times 1 is 3, and so on. 2 times 2 is 4, and 2 times 3 is 6. But once we have 6, we throw away 5 because we're working mod 5, so that just becomes 1. 2 times 4 is 8; again, if we throw away 5, we get 3. Similarly, 3 times 2 is 1, and 4 times 2 is 3. How about 3 times 3? Well, that will be 9;
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we throw away 5, and we get 4. 3 times 4 is 12; if we throw away 10 (just a multiple of 5), then we get 2. Finally, 4 times 4 is 16, which is congruent to 1 after we throw away 15.
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This is a multiplication table. It's not like an ordinary multiplication table that you learn in primary school. In fact, it's sort of similar because the numbers are all pretty small, but it still has a lot of the properties of a multiplication table. The key properties that make it through are, first of all, that the operation is closed. The multiplication, which is the operation, is closed. That means when we multiply two elements,
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we get another element in the set; we're not going outside the set. Secondly, there is an identity element. In this case, the identity element is 1.
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What does the identity element do? It has the property that 1 times anything is the anything, and in either order; it doesn't matter. Another property is that each element, say "g", has an inverse "g-inverse", which has the property that if you multiply "g" times "g-inverse" or "g-inverse" times "g", you get 1. So, for example, the inverse of 2 would be 3 because 2 times 3 is 1, while the inverse of 4 would be—let's look around here for 1—there's
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a 1 in that row right there. So, the inverse of 4 is 4. So, the inverse of 2 is 3, 4's inverse is 4, 1's inverse is 1, and 3's inverse is 2. Those are three important properties, but there's one more which is a little bit harder to check in general unless you have some special information, and that's associativity. Associativity means that the multiplication operation is associative. What's that mean? It means that if you have, say, three elements, G, G1, G2,
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and G3, and if you compute that in that order, it's the same as when you take G1 times G2 times G3. In this example, it's sort of obvious that this is associative because multiplication mod "n" is associative, but in general, that's an important property of the group. So, those are the four properties that define a group, and this is an example of a group. Let's call it G. G is the group that consists of the set plus the operation,
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which involves the set {1, 2, 3, 4} plus the operation encoded in this multiplication table.
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All right, so what Euler used, although he didn't say it in this language,
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was essentially another kind of example. A second kind of example is when we go to a non-prime number like 8 and do the same thing for the numbers that are relatively prime to 8. Remember, those are 1, 3, 5, and 7. If we multiply those mod 8,
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we get another kind of multiplication table that also defines a group, as it turns out. So here, this is multiplication mod 8. Now, what's that look like? Well, 1 times anything is itself, so that's easy enough. Then 3 times 3 is 9, and 9 mod 8 is equal to 1 (we throw away multiples of 8). 3 times 5 is 15, we throw away 8, and we get 7. 3 times 7 is 21, and if we throw away 16 (a multiple of 8), we get 5. This is commutative, so I know that row. I also know that column. 5 times
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5 is 25, and if we throw away 24 (a multiple of 8), we get 1. 5 times 7 is 35; we throw away 32 and get 3. Finally, 7 times 7 is 49; if we throw away 48 (a multiple of 8), we get 1.
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This is another example of a group. Here, there's an identity element. It's clear that each element has an inverse because every row has a 1 again—that's essentially the inverse property. It's sort of obvious that it's associative because we're intentionally doing multiplication mod 8, which is associative. This is another example of a group, but notice that it has some rather different properties than the one before. In this case, each element is its
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own inverse. For example, in this case, 3's inverse is 3, and 5's inverse is 5.
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Although Euler didn't put it in this language, his proof of these theorems was more or less some basic arguments in group theory.
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To understand the next chapter in the story, we have to go to the theory of equations, where group theory raised its head in a very serious way. The theory of equations, specifically polynomial equations, was something that 17th and 18th-century mathematicians were very interested in. In particular, they were interested in formulas like the formula for the quadratic equation or the cubic equation, which had been discovered
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by Cardano. There was also a corresponding formula for a quartic equation (degree 4). People were interested in finding a general formula for an arbitrary polynomial equation of higher degree. This was a major challenge that many people tackled, including Vandermonde and, notably, Lagrange, who wrote a very big and beautiful paper on these things. Eventually, as we mentioned, Galois in some sense understood what was really going on. Galois was the one to
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truly understand that the quintic equation (the equation of degree 5) is not solvable by radicals.
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It's not solvable by radicals. What that means is that if you have a general fifth-degree equation, you know, maybe something like X to the fifth plus 7X cubed plus X minus five equals zero, and you want to solve that equation, then chances are there is no formula that you can write down like the quadratic formula or the cubic formula. There is just no such formula. That's what Galois understood. That was the culminating point: this hope or dream of finding ever more complicated
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formulas for higher-degree equations was just not possible. It stopped at degree four. You could do it with degree three and four, but for degree five and higher, you couldn't. Galois's real understanding of this was built on the work of Lagrange and others. Ultimately, it became clear that group theory was really at the heart of Galois theory. Lagrange was interested in roots of equations, the zeros of equations, and he was also interested in permutations of roots.
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So, for example, suppose an equation has five zeros. Lagrange was interested in what happens if you permute these zeros and various algebraic combinations of these permutations. Group theory really comes in understanding the permutations of the zeros of polynomial equations. We will discuss more about that in a subsequent lecture on Galois theory, where group theory ends up playing a very big role. This involves groups of permutations, which play a central role.
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Lagrange probably anticipated this, but Galois really understood and clarified it. The story continues beyond this. There were other important developments in the 19th century. Gauss had a theory of multiplication of, let's say, quadratic forms, which is a bit complicated but also ended up being essentially group-theoretical. So, Gauss also had the group concept and was working with it to a certain extent. Then, in the 19th century, once Galois's understanding came to
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light and people started to appreciate his work after his untimely early death, many other people started investigating aspects of group theory. These included the famous French mathematician Cauchy, Jordan, the Norwegian mathematician Niels Abel, the prolific English mathematician Cayley, and Klein and his friend and student, Dickson, and undoubtedly others. These were all important 19th-century mathematicians who contributed to the development of group theory.
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What became clearer through their work is that groups actually play a much wider role than these first examples suggest. Groups, in fact, are intimately connected with symmetries and geometries. Klein's very important insight was that different kinds of geometries studied in the 19th century were intimately connected with groups. This connection between these various things was the starting point of a lot of 20th-century development and the
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interest of physicists. Physicists are very interested in symmetries, and it turned out that quantum mechanics and relativity ended up relying on group theory in a very big way. All right, so I want to go back now to a more elementary aspect that is very important: the connection with permutations. Let's have a look at groups and permutations.
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Okay, so there's a basic fact due to Levi ben Gershon, a Jewish scholar from 1321, who realized that there are "n factorial" permutations of "n" objects. So, we can count the number of permutations: if you have "n" things sitting on the table and you want to know how many ways you can rearrange them, there are "n factorial" ways of rearranging them. In the 19th century, it became clear that these "n factorial" permutations actually
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formed an algebraic structure. It's not just that this is a set of permutations, but these permutations can be multiplied in a natural way. When you do, you get a very important group called the symmetric group. The symmetric group, S_n, is just the set of permutations of "n" things. That's the set, and the operation is composition: you perform one permutation and then you perform another permutation. This
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is a quite different kind of group in many ways than the examples we were talking about because, in particular, this is a non-commutative group. In the examples I gave you from number theory, the groups were commutative, meaning "a times b" is "b times a." That's not a necessary property of a group, and so the symmetric groups are examples of non-commutative groups. All right, so to try to set the stage here, I'm going to just look at the
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simplest one. We're going to look at just S_3, which is also pretty interesting and sheds a lot of light on the general situation. If you understand S_3, it's not so hard to imagine how the higher groups work. So, how many permutations of three things are there? There are six permutations because three factorial is six. We can list these permutations in a number of different ways. Let me do that in a kind of a row here.
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So, let's look at a naive way of thinking about it. I'm just going to put 1, 2, 3. I'll write 1, 2, 3 six times, one for each permutation. 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Okay, so this first one is the identity that sends 1 to 1, 2 to 2, and 3 to 3. The second one
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interchanges 1 and 2 but leaves 3 where it is. The third one interchanges 2 and 3 but leaves 1 where it is. The fourth one interchanges 1 and 3 but leaves 2 where it is. Then there are two more permutations: we can send 1 to 2, 2 to 3, and 3 to 1, or we can send 1 to 3, 3 to 2, and 2 to 1.
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Another way of writing these permutations is by specifying what actually happens to the three elements 1, 2, and 3 when you perform the permutation. So, we have: 1 -> 1, 2 -> 2, 3 -> 3 1 -> 2, 2 -> 1, 3 -> 3 1 -> 1, 2 -> 3, 3 -> 2 1 -> 3, 2 -> 2, 3 -> 1 1 -> 2, 2 -> 3, 3 -> 1 1 -> 3, 2 -> 1, 3 -> 2
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There's another useful way to represent these permutations called cycle notation. In cycle notation, we write: (1) (1 2) (2 3) (1 3) (1 2 3) (1 3 2)
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So, how are we supposed to interpret this? The cycle (1 2) means that number 1 gets sent to number 2, and 2 gets sent to 1. Similarly, (2 3) means that 2 gets sent to 3, and 3 gets sent to 2. The cycle (1 2 3) means that 1 gets sent to 2, 2 gets sent to 3, and 3 gets sent to 1. And (1 3 2) means that 1 gets sent to 3, 3 gets sent to 2, and 2 gets sent to 1. Now, let's do some multiplication. Let's agree that our multiplication will be as follows:
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if we take, say, G times H, it means we do G first and then H. So, what would (1 2) times (2 3) be? Here's how you multiply in cycle notation: we are going to perform (1 2) first and then (2 3). Imagine having three things called 1, 2, and 3 in certain positions.
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Let's see what happens to 1. Under (1 2), 1 gets sent to the number 2 position. Under (2 3), 2 gets sent to the 3 position. So, the total effect is that the thing in the first position gets sent to the third position. We put a 3 there.
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Next, let's see what happens to 3. Under (1 2), 3 stays in the third position. Under (2 3), 3 gets sent to the 2 position. So, the total effect on 3 is to send it to 2. Finally, let's see what happens to 2. Under (1 2), 2 gets sent to 1. Under (2 3), 1 is not moved by (2 3). So, 2 is sent to 1. Hence, the result of multiplying (1 2) by (2 3) is the cycle (1 3 2).
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Alright, so I'll illustrate that in another way. If I start with "1, 2, 3" and I perform the permutation "1, 2," I'm swapping these elements around. Then, if I perform "2, 3," I'm swapping these ones around, so 1 gets sent to the third position, 3 gets sent to the second position, and 2 is sent to the first position. The effect is that 1 is sent to the third position, 2 is sent to the first position, and 3 is sent to the second position.
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Okay, so now we can make a multiplication table even bigger and fancier than the one we've had before. Alright, this is a multiplication table in S3. We have the identity, we have (1 2), we have (2 3), (1 3), (1 2 3), and (1 3 2). The same elements go down the column: (1 2), (2 3), (1 3), (1 2 3), and (1 3 2). The identity element is like the identity when you multiply it by anything; you
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just get whatever it is that you're starting with. So that row is easy, and that column is also easy.
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Something else that's easy is if you take (1 2) and you multiply by (1 2). If you perform that transposition twice, you get back to where you started, so that's the identity. Similarly, this is the identity, and similarly, it's the identity. The one that we just calculated over here, (1 2) times (2 3), will be written here. So, here's (1 2) and (2 3), and we computed that as (1 3 2).
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Now we need to compute similarly, so this becomes (1 2 3), (1 3) on the left, (2 3), and the corresponding column here would be (1 2 3), (1 3 2), (2 3), and (1 3). Alright, let me see if I just write this thing down. So, there's the multiplication table for the group called S3. We can check that it's a group; at least from the table, we can see that the e is really the identity element, that the inverse of every element exists because there's an e in
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every row and every column. In fact, notice that every row and column has all six elements with no duplication on any row or any column. That's another of the group properties. The elements (1 2), (2 3), and (1 3) are their own inverses, but these last two are each other's inverses. Now there's an important subgroup of this group. What's a subgroup? A subgroup is a smaller subset that is contained inside the big group but is closed under the operations.
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So S3 has subgroups. First of all, just the identity element itself is a subgroup, trivially, and the whole group itself is also considered a subgroup. There are also subgroups with just two elements: {e, (1 2)}, {e, (1 3)}, and {e, (2 3)}. But there's one other subgroup which is the most interesting one, and that's the one that doesn't involve the transpositions but involves these cycles here. This is actually called A3; it has a name. It's {e, (1 2 3), (1 3 2)}.
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A3 is called the alternating group,
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and it consists of those permutations which are called even permutations.
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What's an even permutation? There are a number of ways to explain what that is. Maybe the nicest way, if you know a little bit of linear algebra, is to say in terms of a determinant.
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If you have an operation like (1 2 3), which sends 1 to 2, 2 to 3, and 3 to 1, if we think of this in terms of having a sort of vector space approach and think of these as vectors, then we can associate this to a matrix. For example, consider the matrix:
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[0 1 0 0 0 1 1 0 0] If you multiply this matrix by the row vector [x y z], then the effect of that is to get the row vector [y z x].
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So that matrix performs the same kind of operation. The first position has actually gone to the third position. For (1 3 2), if I wanted to correspond (1 2 3) in the system, I would do things this way. You can also associate a matrix to the transposition (1 2), which would correspond to the matrix: [0 1 0 0 0 1 1 0 0]
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In this way, we could have another picture of S3, where the S3 are just the permutation matrices, the 3x3 matrices with zeros and ones such that there's exactly one "1" in each row and column. This is like the number of ways of placing rooks on a chessboard so that none of them can capture each other. When you think about S3 that way, then An is easy to say what it is. An is the subgroup of matrices with determinant 1. An are those
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matrices whose determinants are 1. This thing has determinant -1; these two have determinant 1.
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Alright, so that's a very brief introduction to the symmetric group. Of course, we can consider those things more generally, and subgroups play an important role in the theory. There's a very lovely theorem, a fundamental theorem in the subject, Lagrange's theorem, that states if H is a subgroup of the group G, then the order of H (which means the number of elements in H) divides the order of G. So in our example, we had a group of order six. The symmetric group was of order six,
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and the subgroups we found are all of orders one, two, and three, not six. There’s no subgroup of order four inside the symmetric group of order six. Lagrange's theorem says that whenever you have a subgroup of another group, the order of H has to divide the order of G. In fact, that's actually sort of really the reason why Fermat's and Euler's theorems work once you think about it from the group theoretic point of view.
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Alright, so that's a bit of an introduction to the permutation side. Before I finish off with the symmetric group, I'll tell you another very important theorem due to our friend Cayley, which shows that the symmetric group is a fundamental group. Every group, and I mean every finite group G, is isomorphic to a subgroup of some symmetric group. Okay, that's in group theory language. It means equal to. What we're essentially saying is isomorphic to a subgroup of a symmetric group
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SN for some N. So in some sense, if you understand the symmetric groups and their various subgroups, you understand all finite groups. Turns out that understanding all finite groups is an enormous task, and it's still a task that we are far from understanding. Now, in the beginning of the 21st century, people have been working very hard on it, but certainly certain aspects of this are now much more well-understood. But a huge amount of work has gone into the study of, you know,
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what does the family of all finite groups look like and a lot of surprising connections to a lot of other things. In particular, there's a story now of connections with the polyhedra.
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Alright, so in the later part of the 19th century, it became clear that this notion of group was very flexible and allowed us to think more efficiently about symmetry. And so, we started to be able to talk about and use the language of symmetry in a more mathematical proof way than previously. And the study these Philippe polyhedral groups is particularly rich and easy to explain. Okay, so you all know that there are five Platonic solids, right? There are five Platonic solids.
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These are the regular solids whose faces are regular polygons. Here is the tetrahedron, and then there's a cube, and the octahedron, and then there's this dodecahedron which has pentagonal sides, as 12 of them, and then if you join the centers of the faces of these ones you get the dual one, it's called the icosahedron. So those five are the five Platonic solids. And once people started thinking about group theory, they started thinking about the symmetries
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of these things. And Felix Klein, who was a prominent advocate of the importance of group theory in thinking about geometry and also was a foremost educator in the late 19th century, wrote this important little book called "The Icosahedron". It's a book all about icosahedra, and I just want to quote a paragraph from this book. He's talking about rotation. So he's interested in a Platonic solid like this, and he's interested in the question of one of the possible
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rotations that you can do that will keep the thing looking in exactly the same position as it was. Alright, so if I go like this and you close your eyes and I move it, you can open your eyes, you can see that I've moved it, but if you open it and close your eyes and I rotate it like this, then it's gone, gone from itself to itself but it's in the same position overall, you can't tell that I've moved it if you ignore these little things sticking out. Well, maybe it's
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simpler to see in the case of the tetrahedron, right? If you look at this tetrahedron like this and you take a photograph and then I go like this and say, take another photograph, it looks exactly the same. I've moved it but it looks exactly the same. So even we've done a rotation, basically a rotation of three-dimensional space which preserves the solid. So he was interested in these rotations and so he writes, "As soon as we enter upon the task of studying the rotations by
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which the configurations which we have mentioned are transformed into themselves, we are being held to take into account the important and comprehensive theory which has been principally established by the pioneering works of Galois and which we turn the group theory. Originally sprung from the theory of equations and having a corresponding relation with the permutations of any kind of elements. This theory includes, as has long been recognized, every crest, every question
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with which we are concerned in the case of a closed manifold. Now, of any kind of operations, we say of any operations that they form a group if any two of the operations compounded again produce an operation included amongst those first given. In this sense, we have at the outset the proposition that the rotations which bring one of the regular solids into coincidence with itself collectively form a group." So he's identifying that actually associated to each Platonic solid,
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there is a group of symmetries. So each solid has a group of symmetries. Alright, so let's say a little few words about those. Let's start with the tetrahedron. One of the possible symmetries of a tetrahedron, well you notice it has four vertices. So if I'm going to move this around, let's say I fix one vertex, okay? I've got four possible choices of vertices to send that to. I can leave it where it is or I can move it up there or there or there,
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so it's four possibilities. I choose one. So if I'm going to require that this vertex stays in that position, then I still have some more choice about the bottom triangle. I still have three positions that I can move that bottom triangle into. It's basically an S3, they're permutations of the triangle but just the three rotations. So this four times three is twelve permutations altogether. So the rotations or the group of the group of symmetries of rotations,
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let's say, of the tetrahedron, turn out to be A4. Okay, that's inside S4. So S4 is a symmetric group on four letters that has 24 elements, and A4 which is the subgroup of even permutations, is A4, and that's exactly group theoretically the same as the group of rotations of the tetrahedron, the group of rotations for the cube and for the octahedron, which are dual, is S4. There are 24 possible rotations on the cube. We can see that in a similar kind of way if I take out a fixed
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corner and I'm going to rotate the whole cube, well then I have eight possible choices. These are eight vertices, eight possibilities for that first vertex to go to, and once I have that first vertex fixed then there are essentially three more choices because there's sort of three edges emanating from that vertex wherever it is, and I can rotate those in one of three positions just like I did the tetrahedron. So there's eight times three which is 24 positions,
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and the group of symmetries turns out to be a group of 24 elements. But more precisely, it turns out to be exactly the same as S4. In other words, if you write down the multiplication table, the big multiplication table, it would be exactly the same multiplication table that S4 would look like, have the same pattern as for multiplication. Klein and others realize that there's a reason for that,
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and the reason is because of the diagonals of the cube. If you take two opposite vertices of the cube, and that's a large diagonal, how many large diagonals are there? Well, there's that one, there's that one, there's that one. Exactly, there are four large diagonals, and whenever you permute the cube, those four diagonals are being permuted round. So, it turns out that if you label those four diagonals 1, 2, 3, 4, and you perform any rotation
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of the cube, then you're going to get associated to that, I wrote a permutation of the 4 diagonals, and that's then a way of going from a rotation of the cube to a permutation of 4 things. And you have to think about it for a little while, it's a good exercise, great little exercise in fact, to convince yourself that any permutation of the diagonals actually comes from a rotation. Okay, but you're probably mostly interested in what happens for the most beautiful Platonic solids,
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which are the icosahedron, the subject of Klein's book, and the dodecahedron.
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Here's a dodecahedron. Let's see if we can count how many symmetries it has. Okay, well, we try the same thing for smug to figure out how many vertices it has. Okay, I said it has 12 faces, it has 20 vertices. There's 5 at the bottom here, 5 at the top, does this kind of ring of 10 around the middle, just 20 vertices, this one has 20 vertices. Alright, so I'm going to rotate this thing around, let me just do a rotation for you, okay? So let's say I take this one here,
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okay, see if I can move it back, okay, so I'm going to take this one here, and I'm going to send it to there. Okay, since it's in the same position pretty well as it was before if you ignore these little appendages. So if I start with a fixed vertex, there are 20 possible vertices that it can be sent to. Okay, and once I've chosen the vertex where it can be sent to, say right there, how many more possibilities are there? Well, there's three things coming out here,
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so like I could also do this one, this one, or keep it where it is. So there are 20 times 3, 20 times 3 equals 60 possible rotations of the dodecahedron. In fact, turns out it's the same for the icosahedron, because any symmetry of the dodecahedron is a symmetry of icosahedron too, the icosahedron is obtained by taking the centers of the faces of this one that forms my cozy. So what is this group abstractly? Well, it turns out that this group is a high, the symmetric group on five
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letters, has five factorial elements or 120, the subgroup of even permutations inside that is A5, that has half of them, 260. So it turns out that this alternating group is the group of symmetries of the icosahedron or the dodecahedron. So similar to the way the A4 is the group of symmetries to the tetrahedron, this group A5 turns out to be a very important group in the theory because it turns out to be a simple group. It's a simple example of a simple group. I'll tell you a
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little bit more about what a simple group is in a further lecture. Okay, so I might end just by talking a little to say a few more words about Felix Klein who, in a very important lecture at the University of Erlangen, set out what he called the Erlangen Program. And basically, the thrust of this was that, have geometry or a geometry, he is really determined by a group of symmetries. Okay, and so, for example, if we just in planar Euclidean geometry, associated to that,
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or the group of symmetries like rotations, translations, dilations, and so on. And so, this program really established the connection between geometry and symmetries that would play such a very big role in 20th-century both mathematics and physics. So there's a very kind of important conceptual step for people to say, "All right, instead of studying something like this, let's study an object, let's study the group of symmetries on the object." It's a very late
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19th-century such a point of view that the group of symmetries is somehow more intrinsic or more fundamental than the object itself that would turn out to be a very important idea in 20th-century physics. All right, so in our next video, we're going to look at an aspect of this that I haven't mentioned too much about, which is the theory of Galois detection with theory equations, quintic equation, why doesn't it have a formula for extinction. So see you.
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