 So this lecture is part of an online mathematics course on group theory and will mostly be about wreath products. So let's start with motivating them. We've done the groups of order up to 16. So let's have a look at the groups of order 17, 18 and 19. Well, groups of order 17 and 19 are not terribly interesting because these are primes and so the only groups are cyclic. So let's look at groups of order 18. Well, by the Seeloth theorems, G has subgroup of order nine and this subgroup must be normal because it is index two. So G is a semi-direct product. It's of the form a group of order nine, semi-direct product, a group of order two, which must be Z modulo two Z. So we just have to run through the possibilities. So this group of order nine must either be Z modulo nine Z or Z modulo three Z times Z modulo three Z. And for this one, there are only two ways a group of order two can act on a group of order nine. It either acts as one or minus one and in the two cases we get the cyclic group Z modulo 18 Z and in the case when this group acts as minus one we get the dihedral group on 18 points. So we get nothing new there. For this group, we need to look at all the ways that the group Z modulo two Z can act on it. And what we do is we think of this as being a vector space over a field with three elements. And the non-trivial element of this group would be an order two linear transformation. Let's call it sigma. So sigma squared is equal to one. And we can split up this vector space. Let's call this vector space V. V is the sum of eigenspaces of this element of order two. So it's a direct sum of V plus and V minus where sigma equals one on V plus and sigma equals minus one on V minus. And V plus can have dimension two, one or zero. So sigma looks like, well, it either looks like a matrix plus one or it looks like a matrix plus one minus one that's a V plus as dimension one and V minus this dimension minus one or it looks like this. So it acts as minus one everywhere. So these are the three conjugacy classes of linear transformations of order two of this vector space. So this gives us three possibilities for what this group of order 18 is. So this one gives us nothing very interesting. It just gives us Z modulo three Z times Z modulo three Z times Z modulo two Z. This one gives us Z modulo three Z times. And then we've got a group of order three on which sigma is acting as minus one. So we get just the group S three. And this one gives us a group Z modulo three Z times Z modulo three Z semi direct product Z modulo two Z with Z modulo two Z acting as minus one. So this group is a Frobenius group and we'll be talking about it briefly next lecture. This group is an example of something called a wreath product. So I'll say a bit more about that. So what is a wreath product? So a wreath product of two groups G and H which is sometimes been noted by D with a sort of funny wiggly sign followed by H is given as follows. What do we do is we take several products of copies of G and we take these H times. We think of these as being indexed by H and H acts on this by, you can think of H as acting by the regular representation of H in other words, H is just acting on itself by left translations. So we've got a group actor on by H and we can just take its semi direct product. And this is the wreath product of G and H. So what does that have to do with the group of order 18 we had earlier? Well, if we take Z modulo three Z as G and H as a group of order two, we would take this group here, sorry that three Z and then we take a semi direct product of this by group of order two. Now this group of order two acts by flipping these two elements. So you see it's got one eigenvalue plus one and one eigenvalue minus one if we think of this as a vector space. So this is exactly this group we had here. It's one of the three semi direct products of this group by this group. And you can picture this group as, well, if you think of one of these sort of mobiles you have hanging from the ceiling, you might have the ceiling coming down and then it branches into two mobiles and each of these comes down and at the bottom of each of these you have a little triangular mobile that can spin around. So on the bottom of this one, I also think of a little triangular mobile that can spin around and there might be things hanging on the end of these. And now you ask how many ways can you spin this mobile? Well, you can rotate this bit by one third or two thirds of a revolution. So we get a group Z modulo three Z acting on this. And similarly, we get a group Z modulo three Z because we can spin around this triangle but you can also swap around these two things here. So we get a product of these two groups because you can twiddle each of these bits but you can also swap these two things and tie so we get a semi direct product with Z modulo two Z on top of it. So reef products sort of look like automorphism groups of those mobiles you see hanging from the ceiling. At least if you construct your mobile in a careful way. This isn't the first reef product we've had. The dihedral group of order eight is also a reef product. It's the reef product of Z modulo two Z with Z modulo two Z which is just Z modulo two Z times Z modulo two Z with the semi direct product Z modulo two Z sitting on top of it. So if we think of this as a mobile and what we've got is hanging from the ceiling we've got two things sticking out and then each of these has two more things coming down. So D eight is actually a group of automorphisms of this mobile or graph here. We can twiddle those two and twiddle those two or you can swap these two. Or if you want to think of D eight as being the group of automorphisms but square the Z two cross C two just consists of well, this C two might be swapping the square like that. This C two might be swapping the square like this. And the Z two here might be just swapping to the diagonals which you can see swaps around these two automorphisms. In general, this is a sort of tree and automorphism groups of trees quite often turn out to be wreath products. For example, I suppose I take this tree and ask what is the group of automorphisms of this graph here? Well, you can see that I can fix this point and I could permute these four points here. So that would give me an S four and I've got another S four here and I've got another S four here where I can just permute these but I can also permute these three points here and permuting these three points here gives me an S three. And if you put everything together, what you see we have is S four times S four times S four. And then we have a sort of semi direct product with S three on top of it. So this is just a wreath product S four wreath S three. And obviously you can have more complicated trees like that. So another way wreath products turn up is in the seal of subgroups of symmetric groups. So I think the best way to illustrate this is just to have an example. Let's look at the seal of three subgroups of S 27. So what do these look like? Well, I'm going to draw the following picture. I'm going to draw a sort of tree. I'm going to have three things coming from each node. And I'm going to have three things coming down here. And you see down here I have 27 points. So I can label them one, two, three, four, five, six and you can figure out what the rest of it are for yourself. And now I'm going to write down the seal of three subgroup. First of all, I'm going to do a cyclic permutation of these three points. I'm going to take that in my seal of three subgroup and I'm going to like put cyclic permutation of these three and the cyclic permutation of these three. So these generate a group isomorphic to Z modulo three Z cubed. And now I'm going to allow myself to do a cyclic permutation of these three branches here. So I might map that to that to that and to that. And I would carefully fix the orders down here. So I'd map one to four and two to five and three to six if I was mapping this to this and so on. So this gives me another element of order three. And this element of order three is going to be one, four, seven, two, five, eight, three, six, nine. So you can see that this element together with these three elements is now giving me a group Z modulo three times Z modulo three times Z modulo three. Semi direct product Z modulo three, which is Z modulo three Z, Reath Z modulo three Z. And now I can do exactly the same thing here. So I get another group, which is Z modulo three Z, Reath Z modulo three Z. And of course I get the same thing here. Get a Z modulo three Z, Reath Z modulo three Z. And you can see that I'm getting yet another copy of Z modulo three Z, which just permutes these three things here and it permutes all these three groups. So if I take this element and all these three groups here, what I'm getting is a group Z modulo three Z, Reath Z modulo three Z. And I get three of these and I have a group Z modulo three Z sort of acting on top of them. So I get a group Reath Z modulo three Z. So I get this sort of tower of Reath products where I take a Reath product for two Z threes and then take a Reath product of that with Z modulo three Z. And it's pretty obvious that this works if you replace Z by any prime and you can obviously get bigger and bigger towers of trees. If the seal of subgroup has ordered a power of P, then you get a single tree like this. In general, the seal of subgroups of symmetric groups tend to be products of Reath products. For example, if we look at the seal of two subgroup, let's just do the seal of two subgroup of S 13. So what I do to work this out is first of all, I draw this tree here, which is eight points. And then I have another tree with four points and then I have another tree with one point. So we're writing 13 equals two cubed plus two squared plus two to the zero. So I get a tree of height three and a tree of height two and a tree of height zero. And from this, I get a Reath product Z modulo two Z. Reath Z modulo two Z. And from this tree, I get a group Z modulo two Z Reath Z modulo two Z. And from this group, I get a Reath product of no groups whatsoever. So I just get a group one. So you see that the seal of two subgroup of S 13 is going to be this Reath product times this Reath product times this Reath product. And the seal of subgroups of all other symmetric groups look kind of similar. You can just build them up by taking these incredible Reath products. Incidentally, any P subgroup is contained in some symmetric groups. So it must be contained in one of these Reath products. And this sort of might give you the idea that maybe you could classify P groups just by finding all subgroups of these Towers of Reath products. So if I take a Reath product of lots of copies of Z modulo PZ, all you have to do to classify Reath groups is to find all subgroups of that group. Unfortunately, this is completely hopeless because subgroups of Towers of Reath products are incredibly complicated and practically impossible to enumerate. So that doesn't really help. There's also a way that Reath products sort of turn up in physics. If we look at the Reath product G twiddle H, I don't know what this sign is called. You can see it's G times G times G Reath, sorry, semi-direct product H. Well, there's a variation of this. Suppose H acts on S. Suppose H acts on a set S. Then we can form G times G times G and so on where we have S copies of G. And since G acts on S and we index these by S, H acts on this. So we can again form a semi-direct product of this with H. And this is also sometimes called a Reath product. It's a slightly more complicated one where instead of H acting on itself, it's acting on some other set. Now you can think of this set here as being G to the S Reath H, where G to the S means all functions from S to G. Now we can also do this where S might be a topological space and G might have a topology. So we could take it to be all, say, continuous functions. So if G and S both had topologies, then it would be silly to take all functions. You might want to restrict to continuous functions or maybe smooth functions or something like that. And there's a construction very similar to this that turns off in physics. So here what we do is we take S to be spacetime and we take H to be symmetries of spacetime, which is, it might be the Poincaré group or the Lorentz group or whatever. And we might take G to be some sort of gauge group. For example, if you're doing electrodynamics, you might take G to be the circle group S1. And then G to the S will just be all functions from spacetime to a circle. That's all nice functions. It might mean continuous or smooth or whatever. I don't care. And this is also sometimes called a gauge group. This is an infinite dimensional group, which can be thought of as a big group of symmetries that controls electrodynamics. And then you can form the group G to the S and take a semi-direct product of it with the Poincaré group. And this large group here then also acts as a big group of symmetries on quantum electrodynamics in some sense. So these large groups, this is more or less a wreath product of G with the Poincaré group where this wreath product, you're using the action of Poincaré group on spacetime and so on. So these wreath products do sort of turn up in physics when you're doing quantum field theory. More generally, if you've done fiber bundles, you can guess that you can get similar groups for any fiber bundle over a manifold, except this won't be functions from S to G. It will be sections of a fiber bundle or something like that. Okay, so next lecture, we will be looking at groups of order 20 and Frobenius groups.