 So this is a mathematics lecture on group theory, more precisely about Cayley's theorem. So in the previous lecture, we had two ways to think about a group. First of all, it could be the symmetries of an object. So that's a sort of concrete way of viewing it. On the other hand, we had an abstract way of viewing it where it was a set with a multiplication satisfying associativity and the existence of an identity and the existence of inverses. And we showed that if you've got the symmetries of an object, then we can get an abstract group where the multiplication is just composition of symmetries and these axioms are all very obvious. And we can ask the converse questions. But as we've got an abstract group satisfying these axioms, is it the symmetries of some object? Or in other words, how have we actually found all the axioms for a group that we need? So, and this is what we're going to discuss this lecture. So Cayley's theorem says, yes, that if you've got an abstract group, then there's some object that it's the symmetries of an object. So Cayley's theorem is essentially this arrow here saying that the abstract definition of a group that we've got really is the same as saying that it's just the symmetries of some object. So to do this, we just need to define what we mean by a group acting on a set. So we say a group G acts on a set S. This means we're given a function from G times S to S. And this is usually written, you take an element G and an element S of S, and you write this as G of S. And it has to satisfy the following rules. G times H of S equals G H of S and one of S equals S. And these are just the rules you get if you think of G as being the symmetries of an object S. And then this just says that composition of functions is, well, this is the definition of composition of functions. And this just says the identity element really acts as the identity on S. So a fairly typical example, you might take G to be the symmetries of say an octahedron. And you might take S to be say the set of six vertices of the octahedron. So G has 24 elements. And in this case, S has six elements. And there's a map from G times S to S, which just gives the action of the symmetry on one of the six vertices. Well, if we look at this rule here, you notice this looks very much like the associative rule for multiplication, except that S isn't in G, it's in some other element S. And from this, we notice that there is an obvious action of G on itself. So G acts on S equals G. So I'm going to take the set S to be the same as G. And we're just going to define the action of G on an element S just to be the group multiplication of G and S. So this is the group product. And this is the definition of the action of G on S, which is equal to G, but I'm writing it with a different letter S because we're thinking of, if I write S, it means we're thinking of G as being just at a set of points and forgetting about the group multiplication. And what this means is that G is a subgroup of the permutations of S. So the permutations of S is a group, it's just the group of all bijections from the set S to itself. And what does a subgroup mean? Well, G is a subgroup of H, means G is a subset of H with the same product and containing one and closed under the group product. So it's just like a, it means G is a group and it's contained in H and the product in G is the same as the product in H. So this is a sort of weak version of Cayley's theorem. It says that any group is a subgroup of a group of permutations of some set, but it doesn't yet give G as being the group of symmetries of something because the group of all symmetries of S is the group of all permutations and we've only got some of them. So to get a more precise version of Cayley's theorem and get G to be exactly the group of symmetries of something, what we need to do is to add some extra structure. And what we defined earlier was a left action of G on S means we've got a map from G times S to S taking G and S to G S. Well, we also wanted to find a right action. This is going to be a function of S times G to S taking S and G to an element S G. It's a bit difficult to write right actions because we're used to writing functions on the left of the thing they act on. Sometimes it's useful to write functions acting on the right of the things they act on and the trouble is when you do this notation it gets to be a bit of a mess. And of course it should satisfy the obvious actions that S times G H is equal to S of G H and S of one is equal to S. So it's just like a left action except it's the other way round. By the way, left actions aren't necessarily the same as right actions. So the point, so for example, we have a right action of G on itself and this isn't necessarily the same as the left action of G on itself. So the question is, is G of S equal to S of G? If S and G are in the group. And when S of G equals G S, we say S G commute. This is actually rather similar to the word commute when you go off to San Francisco or whatever for your job. Commute is just a variation of the word change. You know, a mute is as in mutation meaning a change of something. And here commute means you exchange two things. So we exchange S and G and similarly when you commute to work you're changing your position. So here G is sort of commuting to the other side of S. So if any two elements of a group commute then we say G is commutative or sometimes abelian because this is named after the great Norwegian mathematician Arbell who studied commutative groups or rather abelian varieties. So anyway, the problem is that in general groups don't commute. The simplest example of this is a triangle and we take the group of its symmetries and we can have a blue symmetry which is just reflection in this line. So we exchange these two points here or we can have a red symmetry which just rotates it. So let's call this red symmetry B and this blue symmetry A and we better number the points one, two and three. And we use this, it's a bit tiresome writing down functions. So what we do is we use a compressed notation for functions, we write A equals one, two where this means the function taking one to two and two to one. And similarly we write B as one, two, three and this is an abbreviated form of the function taking one to two, two to three and three to one. And let's work out what AB is. So if we multiply A by B it means we first do B and then A. So it takes one, well B takes it to two and then A takes it back again. So it fixes one, whereas it takes AB takes two to three and then A fixes three and similarly you can see it takes three to two. So AB is two, three and then you can similarly work out the element BA is equal to one, three and these are just definitely different. So this is the smallest example of a group that is non commutative. So the right action of the group on itself is definitely different from the left action which is why we need to be so picky about whether we're acting on the right or the left. Anyway, we notice that suppose G acts on S which is equal to G on the left. Then this preserves the right action of G. What this means is that G acting on S H is equal to G acting on S multiplied by H. In other words, acting on G with the left commutes with the action of G on the right. So it sort of preserves the right action. And you notice this is just the associative law. So although two elements of G acting on the left need not commute with each other and two elements of G both acting on the right need not commute with each other. If you act with G on the left and H on the right it doesn't matter which order you do then you can multiply by H on the right first or you can multiply by G on the left first and you get the same answer. And now we can treat G as being the symmetries of an object with structure. So this is what we want to do. We want to show there's a mathematical object with some structure such that G is all its symmetries. Well, this object is just going to be S which is the same as G except you forget that it's a group. The structure is going to be the right action of G on S. And this is really confusing because you notice the group G is appearing three times in different ways. First of all, it's this set S that we're acting on. It's the first way it appears. Secondly, it's acting on the left on S. So these are symmetries acting on the left. And thirdly, it's appearing as something acting on the right on S. And these are all different. I mean, the left action is not the same as the right action and so on. And the left action is the same as the right action but the point is that G is the set of all symmetries of S action on by G. And this result is trivial but rather confusing. So suppose F is a symmetry of this set S preserving the right action. So we want to show that F is multiplication on the left by some element of G. And we just notice that F of S equals F of one times S because S is one times S which is equal to F of one times S because it preserves the right action. So F is just multiplication by the element F one of G. So any symmetry of G with its right action is actually just an element of G. So G is indeed the set of symmetries of some object with its structure. Notice by the way that this action of G does not preserve the group action on S. That would be saying that G of AB is equal to G of A times G of B which is certainly not true in general. So the action of G on S doesn't preserve the group structure of S but it does preserve the right action of G on S. So all this is incredibly confusing so it'd be a good idea to draw some pictures. What you can do is you can draw something called the Cayley graph of a group. And the best way to do this is just to have some examples. So let's take G to be the symmetries of a rectangle. And a rectangle has four symmetries. First of all, we can just have the identity symmetry or we can sort of flip it like this around a horizontal axis or we can flip it around a vertical axis or we can do a 180 degree rotation which involves flipping these two opposite corners. And what I'm going to do is I'm going to indicate the elements of the group by primary colors. I can find the, there we are. And there are four primary colors. You may have heard from scientists that there are three primary colors but if you go to any store for babies you'll notice there are four primary colors, red, blue, yellow and green. And I'm going to use these, the four elements of the group. So what you do is you draw a point for each element of the group. So this is going to be the element one and this is going to be the green one and this is going to be the blue one and this is going to be the yellow element here. And now what I'm going to do is I'm going to draw colored lines between these to indicate what the right action of the group is. Well, if I take the element one and multiply it on the right by this green element I get the green element. And similarly, if I do the green element twice I get back to one. And similarly, you can check that the blue and yellow elements are swapped by this. And similarly, if I multiply the blue element on the right it transforms things like that. And if I multiply by the yellow element it swaps those two. And if I multiply by the red element which is the identity everything just stays the same. So we get like that. So I've got a four points and I've got little colored lines on them with arrows on them indicating what happens if you multiply one of these four elements on the right by an element of the group. So this is called the Cayley graph. And if you think about it the Cayley graph is really just equivalent to the group together with the right action because the right action is given by all these colored lines. So G is the symmetries of this graph. And by the symmetries I just mean a map, a bijection of the vertices that maps every colored line with an arrow to another colored line with an arrow in the same direction. Well, this example is a little bit oversimplified. The trouble is that this group G is actually a billion so the right action is the same as the left action. And you don't see the full confusion of this example. So now let's do a more complicated Cayley graph of a non-commutative group where the right action is different from the left action. So we're gonna take G to be the group of all symmetries of a triangle. So I can mark the vertices of a triangles one, two and three. And now using the notation that I introduced earlier it has six elements, one of which is the identity. And then you can swap the two corners there or two corners there or two corners there. Or you can rotate a third of a revolution clockwise or a third of a revolution anticlockwise. So there are six elements of the group. And now to do the Cayley graph I'm going to need two extra colors. See if I can find the pens there they are. So I'm going to draw six points of different colors like this. So I'm going to red point for the identity. And then I'm going to have a blue point for one, three, two and a green point for one, two, three and another green point for one, two with a different shade of green. And a yellow point for two, three, which I'm going to put down here and a different shade of blue point for the element one, three. And now I'm going to draw arrows between these indicating what happens if you multiply on the right by various elements. So if I multiply the identity on the right by this element one, two, three I get the element one, two, three and you can check it goes around like that. And if you calculate what happens to these ones it does that. And the dark blue is very easy to do because this is just the inverse of the green element. So it goes in the opposite direction. One of my arrows on the green thing the wrong way around. So you do it like that. And then the yellow element if you multiply the identity by the yellow element you get the yellow element. So we swap those two and it was ordered two so the arrows go both ways on these and I won't bother to draw them. And you can check it swaps those ones. And similarly, the light blue element swaps things like that. And the light green element swaps things like that. I hope I've got all the colors right. Honestly, I'm just putting these colors at random and I doubt that anyone would notice the difference. So this is the Cayley graph of the symmetric group on three elements. The six vertices are at six points and these lines with arrows on tell you what happens if you multiply by everything on the right. For instance, whenever we have an element G and an arrow, the element here is G times the element one, three, two. And similarly for all the other colors. So this group of six elements is the group of automorphisms of this mess here. Obviously this is not the simplest thing it's the group of automorphisms of. It's the group of automorphisms of a single triangle which is obviously rather simpler than this but the point is this gives a systematic way of constructing a Cayley graph from any group and the group is then the group of symmetries of the Cayley graph. So to summarize the axioms of a group capture the concept of the symmetries of an object, whatever that means that anything we can prove about all symmetries of objects can be deduced from the axioms of a group. We haven't missed out any vital axioms. The idea that the group can act on itself and either the left or the right is rather confusing. So it suggests there are two ways a group can act on itself. In fact, there are eight natural ways a group can act on itself. And it's incredibly confusing trying to sort these all out. So I'm going to finish this lecture just by describing the eight possible ways a group can act on itself. So these are the eight actions of G on S equals G. As usual, I'll write S as for G when I'm forgetting about the group structure on G and just thinking of S as a set. And there are four left actions and four right actions. So for each left action, I've got to tell you what G of S is. And for each right action, I've got to tell you what S of G is. And there's one left action, which is very easy. I just let G of S be S. So this is the trivial action. And similarly here, I can do that. This is trivial. In other words, G acts on itself just by doing absolutely nothing. Then we have, it can take G of S to G times S which is the obvious one or S times G. This is called left or right translation. So these were the actions we were using. There are another pair of actions. We can let G act on S by taking it to S G to minus one or G to minus one S. And finally, we can sort of combine these and take S to G S G to the minus one or G to the minus one S G. These are called the adjoint actions. And we'll be using them quite a lot later. This one doesn't really have a name. And you may wonder why is this a left action and this a left action, but this a right action and this a right action? Why couldn't we make this into a left action? Well, S times acting on this by making this into a right action doesn't actually give you a group action. So suppose we tried to find a left action like this. So we would say G of S equals S of G. Put a question mark there because we'll see it doesn't work. Well, let's try G H of S. That would have to be S times G H. On the other hand, it would also have to be G of H of S which would be S of H then applied to G. And you can see these are not equal. At least they're not equal if the group is not commutative. If the group is commutative, then these are the same. On the other hand, if we put G of S equals S of G to the minus one, let's see if this works. Well, G H of S will be S G H to the minus one which is equal to S H to the minus one, G to the minus one because when you take inverses you swap the order. And this should be equal to G of H of S which is equal to G S H to the minus one which is equal to S H to the minus one, G to the minus one. And now you're okay because these two are equal. So it's very confusing trying to distinguish between left and right actions and trying to remember whether you should take the inverse of G before acting on it. So this is a sort of table to try and sort out this confusion.