 This lecture is part of an online mathematics course on group theory and will be mostly about nilpotent groups. So we'll start by looking at groups of order 16 because we've classified all the groups of order less than 16. Well, we're not actually going to give a proof of the classification because this is kind of a bit tedious, but let's just list them to see what we get. So we're going to list all the groups of order 16 we can think of. First of all, there are five abelian groups that we considered in the previous lecture and these correspond to the fact you can write 16 as a product of numbers in the following ways, four times two times two, two times two times two times two. So we can get five abelian groups as product of cyclic groups of these orders. Then there are two products of z modulo 2z with either the dihedral group or the quaternion group of order 8. Then there are several groups. There are four non-abelian groups with an element of order 8 and these are as follows. We get a generalized quaternion group. This is the same as the binary dihedral group of order 16, which means you take the dihedral group of order 8 as a group of rotations and take its double cover in the group s3 of quaternions. Then there are three groups that are semi-direct products of z modulo 8z by z modulo 2z. So these are generated by an element a with a to the 8 equals one and an element b with b squared equals one generating these two and then bab to the minus one can be a cubed a to the five or a to the seven. If it's a to the one we just get an abelian group of course. A to the seven is the dihedral group. This one here is called the semi-dihedral group and this one here is just nameless. Next, we have various semi-direct products of various groups and these semi-direct products are as follows. We've got a group z modulo 4z, semi-direct z modulo 4z with z modulo 4z acting non-trivial on this. We've got a group z modulo 2z times z modulo 2z semi-direct product z modulo 4z and there's a group z modulo 4z times z modulo 2z semi-direct product z modulo 2z and there's frankly not a lot of any great interest to say about many of these groups. We can make some minor comments. This is sometimes called the poorly group and in quantum mechanics we have several matrices called the poorly matrices. Oops, it should be a zero, nought minus i by nought and one minus one and we can also add the identity group and we can take all these four groups and multiply them by power of i and that gives us 16 matrices forming a group and that's this group here. I mentioned earlier that we can sometimes distinguish groups by count of the number of elements of various orders and that fails for this group because if you take, I think it's this group, hope I've got it right, and this group here, these are both non-abelian and they both are three elements of order two and 12 of order four. So you can't distinguish these groups by counting the number of elements and seeing whether they're abelian or not. And this is actually fairly typical of groups of order two to the power of something or for that matter p to the power of something for p odd. You start to get large numbers of groups and it's kind of hard to say anything very interesting about any particular group. The groups of order 16 are all nilpotent. I covered nilpotent groups in an earlier lecture but since I can't assume that anybody either watched or remembered that lecture I just reviewed nilpotent groups. So you remember a group g is called nilpotent if you can kill it by repeatedly killing the center. So we take g nought as a group, as our group we put g1 equals g nought over the center and g2 equals g1 modulo the center of g1 and so on. And if gn is trivial the group is called nilpotent. And we saw earlier that a groups of order p to the n for p prime is always nilpotent. And let's just recall the proof. All conjugacy classes have order a power of p because the size of the conjugacy class is the size of an orbit of a group so it must be the index of some subgroup which is a power of p. So the size is equal to one or is divisible by p. Now the union of all the conjugacy classes is g so the sum of the orders of all conjugacy classes is divisible by p. So the number of conjugacy classes of size one must be divisible by p because it's just the order of g minus all these other conjugacy classes. So the number of conjugacy classes of size one is divisible by p. Well a conjugacy class of size one is just an element of the center where the center is something that commutes with all elements in g. So the center must be non-trivial because its order is divisible by g so you can kill off the center to get a smaller group and then by induction you can keep doing that. We also notice the product of two nilpotent groups is nilpotent. This is very easy so I won't bother giving a proof of it. So we find that any product of p groups is nilpotent so and we're going to show that conversely any finite nilpotent group is a product of p groups. I'm just going to talk about finite groups for most of the rest of this lecture so I'll just implicitly assume they're finite and to show that so we've shown that a product of p groups is nilpotent. To show that a nilpotent group is a product of p groups it's useful to introduce a third condition that all c-love subgroups are normal. This means there's just one c-love p subgroup for any prime p. So what we're going to say is one, two and three are equivalent for finite groups. So this means that for finite groups being nilpotent is almost the same as being a p group because taking a product of groups is a fairly trivial operation. Unfortunately this doesn't really help a lot with studying nilpotent groups because it turns out that p groups are such a mess that the only way to study p groups is to use the fact that they're nilpotent. But anyway so we've proved that condition one implies condition two. So now let's show that two implies three so condition two says that they're nilpotent and condition three says that the c-love subgroups are normal and to do this we're going to use induction on the order of the group. So pick some element g in the center of g with g to the p equals one where p is prime. So we're picking an element of order p in the center and we can do this because the group is nilpotent and we look at g modulo the group generated by this element and g modulo p is still nilpotent so all c-love subgroups of g over p are nilpotent by induction. Here we're using the fact that if you've got a nilpotent group and you're quotient out by something then the result is still nilpotent which is easy to check. And now we look at the q-c-love subgroups of g so suppose q is equal to p then a c-love q subgroup of g over p is normal and the inverse image in g is a c-love q or equals p subgroup of g and the fact that it's normal in here shows that its inverse image is normal. Now suppose q is not equal to p and pick a c-love q subgroup of g over p and we look at the inverse image I'm going to say pick the c-love subgroup of g over p sorry there's only one of them and look at the inverse image in g so the inverse image has order p times q to the n for some n where q to the n is the largest power of q dividing the order of g and this has a center of order at least p because the group of order p is generated by this element in the center and it has a c-love subgroup of order q to the n however this c-love subgroup must be normal because the only conjugates of it would be conjugates of it by elements of the center and the center just fixes any c-love subgroup so the c-love q subgroup is of this group is unique and it must be the c-love q subgroup of g so we've shown by induction if a group is nil potent then all c-love subgroups are normal finally we show that three implies one so you remember one says that it's a product of p groups and three says that c-love subgroups are normal so we want to show that if g has the property that all c-love subgroups are normal then it's a product of p groups and what we do is we pick c-love subgroups p and q so this is going to be a c-love p subgroup some prime p and this is going to be a c-love q subgroup and we're going to take p not equal to q and the main point is that p and q commute meaning every element here I mean every element in the group p commutes with every element in the group q and let's look at this suppose a is in p and b is in q then look at a b a to the minus one b to the minus one and we notice that this is in in q because q is normal so this is in q and similarly a b a to the minus one b to the minus one well this bit is in p because p is normal so this element is also in p so a b a to the minus one b to the minus one is in p is in p and q but p and q the intersection is just trivial because this is a p group and this is a q group and p is not equal to q so there any element in this in in this intersection of order of power of p and order of power of q so it would have to be one so a b equals b a so all the c-love subgroups for different values of p and q commute with each other now we know by the c-love theorems that there is a c-love p subgroup for every prime p um and um these subgroups we've just shown these subgroups all commute with each other and from this it follows very easily that the group is just a product of these p-c-love subgroups so that shows that um a group being nil potent a group being a product of p groups and a group having all its c-love subgroups normal are all equivalent um so next lecture we will move on to groups of order 18 and discuss wreath products