 This lecture is part of an online mathematics course on group theory and will mostly be about Frobenius groups. So as usual, we will start by classifying a few finite groups to see what interesting groups turn up, and we've done groups of order less than 20. So let's look at groups of order 20. And by now it should be fairly obvious what we do. We notice that there is a seal of subgroup of order 5, and the number of seal of subgroups must divide the order of the group and be 1 mod 5. So this subgroup has to be normal. So we see easily that G is a semi-direct product of a cyclic group of order 5 with a group of order 4. Now we just run through all the possibilities and see what, if anything interesting turns up, the automorphism group of a group of order 5 is z over 5z star, which is z over 4z. So now if this group of order 4 is z over 2z times z over 2z, there are two possibilities. First of all, the action might be trivial, in which case we just get a product z over 5z times z over 2z times z over 2z, which is not very exciting. All we get, one of these z over 2z's might act non-trivially on this, and then you can see we just get a dihedral group of order 10 times a group of order 2, which again is nothing to get excited about. So now let's look at what happens if this group of order 4 is z modulo 4z. Well, there are three essentially different ways that a group of order 4 can act on a group of order 5. First of all, it can be trivial, but in this case we just get a product of these two groups, which is cyclic of order 20, again nothing to be excited about, or we could have this acting as a group of order 2 on this group of order 5. So we get a group z over 5z, semi-direct product z over 4z, with a generator of this acting as an element of order 2 on the automorphism group of this. This is actually one we've seen before. It's actually the binary dihedral group of order 20. So you remember this means we take the group of rotations of order 10 given by the dihedral group and take its inverse image in the unit quaternions. So the final example is group z over 5z, wreath product z over 4z, with an element here acting as an order 4 automorphism of this, and this is the so-called AX plus B group. Here it consists of all transformations taking X to AX plus B where A is in z modulo 5z star and B is in z modulo 5z. So you can think of it as being the group of linear affine transformations of a line over the field with five elements. And this is the one that's going to motivate the rest of the lecture today. This is a Frobenius group. So what is a Frobenius group? Well, a Frobenius group G, so Frobenius group G is a group acting transitively on a set such that, first of all, no element of G other than one fixes two elements of S. I should say that fixes at least two elements of S. Secondly, S is not the regular representation. So the second just eliminates the stupid case where G acts by left translations on itself, which obviously satisfies condition one but isn't terribly exciting. So what examples have we seen of Frobenius groups? Well, first of all, we have the AX plus BY group over any field. Here the set S is the elements of the field and the group just consists of all transformations of the form AX plus B where A is in a non-zero element of the field and B is an element of the field. And this group obviously is ordered the number of elements of the field times the number of elements of the field minus one, least if the field is finite. So if the field is ordered two, well, this doesn't quite work because we just get a group of two elements. If F is order three, then we get the group S3 acting on three points. If the field is the field of order four, then we get the group A4. And if the field has order five, we get the group of order 20 that we just mentioned earlier. There have also been several other examples of Frobenius groups. If we take the dihedral group Dn for n odd, this is a Frobenius group. For instance, if we take D10, it's the group of automorphisms of a pentagon. And the set S is just the vertices of a regular pentagon, which or n-gun pentagon in this case, because the dihedral group acts transitively on the vertices. But if you fix two vertices, then that determines everything. If n is even, this fails. For example, if we take a square, then there is a non-trivial automorphism fixing these two vertices because we can just flip the square like that. So if there are two vertices that are diagonally opposite each other as happens for n even, then we don't get a Frobenius group. Another example is the example of the group of order 18 we had earlier. So we took Z over 3Z squared, semi-direct product Z over 2Z, where you remember this acts as minus one on Z modulo 3Z squared. And you can see this is a Frobenius group acting on, say, all the conjugates of this group Z modulo 2Z. So Frobenius groups are actually reasonably common. There are two basic theorems about Frobenius groups, which I'm not going to prove. The first one is by Frobenius. So suppose H is equal to the subgroup fixing a point. And let K be all conjugates. So all elements not in a conjugate of H together with the element one. So all elements not in a conjugate of H. This just means the elements fixing no points of the set S. And Frobenius proved this really rather remarkable theorem that K is a normal subgroup. So the tricky part is to prove that K is a subgroup, which isn't all that obvious. Once you know it's a subgroup, it's trivial to prove that it's normal. And furthermore, G is a semi-direct product of K with H. So this group K is called the Frobenius kernel. So Frobenius proved that the Frobenius group actually has a homomorphism to H with kernel K. And the proof is not all that difficult, but it uses character theory of finite groups, which we haven't actually covered. So I can't give the proof. And there's a much harder theorem due to Thompson, which says the Frobenius kernel K is nilpotent. Well, if you look at all the examples of Frobenius groups we had so far, the Frobenius kernel is not only nilpotent, it's actually abelian. It's quite difficult to find an example of a Frobenius group where the Frobenius kernel is not abelian. One example is the following group here. You take the kernel K to be all matrices of the form 111 nought nought nought star star star. That means there can be anything in gl3 of the field with seven elements. And we take H to be the group generated by one, two, four. So these diagonal elements. Then the group KH is a Frobenius group of order seven cubed times three. So we'll finish off just by doing groups of order 21, 22, and 23. So order 21, these groups are of order PQ of seven times three. And we saw earlier that there are just two possibilities. One is the order 21. And the other is a semi-direct product Z over seven Z, semi-direct product Z over three Z. And the only reason for mentioning it in this lecture is this happens to be another Frobenius group, as you can see. Order 22, there's nothing interesting. This is of the form two P, so it's either dihedral or cyclic. Order 23 is prime, so all groups are cyclic. So that leads us on to groups of order 24, which will be the topic of the next lecture.