 This lecture is part of an online mathematics course on group theory, and in this lecture we're going to be explaining why we're not going to try and classify p-groups. So in previous lectures we've more or less classified all groups of order less than 32, so we can ask what about groups of order 32? And we're not going to classify them because there are 51, and it's really not terribly exciting to look through them all. So what do you do if you want to understand groups of order 32? Well, what you can do is you can look them up in the big red book of groups of order 32, and there really is a big red book of groups of order 32. I have a copy here, and if you look at the little picture of me on the right of the video I'm trying to hold it up so you can read it, and if you look carefully you might be able to see it's by Marshall Hall and James Sienian. It says the groups of order 2 to the n for n less than or equal to 6, and this book is huge. It doesn't actually fit on my bookshelf. So let me show you what's inside it. So what is inside it is they have these amazing diagrams of all the groups of order 1, 2, 4, 8, 16, 32 and 64. So this is a diagram of equivalence classes of subgroups of one of the 267 groups of order 64, and some of these diagrams get really spectacular. For instance there's one here, I can't really, it's so big I have trouble fitting it underneath the document camera, you can probably see most of it there. As I said, this is one of 267 diagrams, although I must admit that's one of the most complicated ones. It gets even worse if you look at two groups of higher order. For instance groups of order 2 to the 10, how many are there? Well the answer turns out to be 49487365422, I guess these were classified by computer rather than classified by hand. So classifying groups of order 2 to the n or for that matter groups of order p to the n for n large is rather like trying to classify all the individual grains of sand on a beach. There are just ridiculously large numbers of them and most of them kind of look pretty similar and it's, they're sort of rather difficult to distinguish. Many of them look much the same. So what I'm going to do this lecture is show why there are so many groups of order of p to the n. So we're going to study groups of order p to the n for n large and try and get some sort of rough idea of why there are so many. And in general we know these groups are nilpotent, so it means they have the following structure. We start with the group G and we take G modulo center and let's call this group G1 and then we take G1 modulo the center and that's going to be G2 and eventually Gn is the trivial group. So in general groups of prime power order may have a long series like this and the longer the series is the more complicated they are. So you may think we're going to construct large numbers of groups by taking a very long complicated series like this but we're not. We're going to look at groups of almost the simplest possible structures. So we're going to take G has a center of the form Z modulo pz to the m. So let's call this the center of G and as I said Z is used for both the center of G and for the integers and I'm sorry about that I can't do anything much about it. It's standard notation for both. So and we're going to take G modulo the center of G to be another group Z over pz to the power of let's not use this letter n let's call this m prime. So both of these are going to be elementary abelian groups that are just going to be product of p groups and the length of this series is just two. So we have one contained in the center of G contained in G and this is an elementary abelian p group and if we quotient it out by it we get a group that's actually a billion of the simplest possible type it's just elementary abelian. So we're going to restrict these very easy groups. So G is an extension Z modulo pz to the m times Z modulo pz the m prime and we can think of this as being a vector space of dimension m prime over the field of p elements and we can pick a sort of basis for this so we have a basis a one up to a m prime and we can look at what is all the elements a i a j a i to the minus one a j to the minus one. So we take the commutator and since this is abelian the commutator must lie in the center so this is in fp to the m for some m and turns out that if you take a basis and choose all these commutators it's not very difficult you can get a group such if you just choose an element here to be that commutator you can define at least one group with these commutator relations in fact you can define others because we haven't actually specified what the pth powers of the a i's are but we won't worry too much about that. So how many ways can we choose that? Well there are about p minus p times p minus one times so p times p minus one over two ways of choosing pairs i j because if we swap i and j we're only going to take the inverse of this so we can really only choose this for i not equal to j. So for each of these sorry that's not a p times p minus one over two it's m prime by m prime minus one over two and for each of these we have an m dimensional vector space of things we can choose so the dimension of all the bilinear maps from z over pz to the m prime times z over pz to the m prime to z over pz to the m is about m prime squared over two times m. So altogether the vector space of all these maps has size about p to the m prime squared times m over two here we're not worrying about this m prime minus one we're just giving very rough estimates for the size of everything. So now we have m plus m prime is equal to n where the order of the group is p to the n so we want to maximize this given that m plus m prime equals n and that's an easy calculus problem we find it's given by m prime is about two-thirds n and m is about a third of n. So altogether we seem to be getting about p to the two-thirds n squared times a third of n over two different groups which is about p to the two over 27 n cubed well in fact we get fewer groups than this because we haven't accounted for symmetries for instance we could have chosen a different basis of the ai's and this might give us two of these this collection of p to the two over 27 n cubed groups that are actually the same well let's sort of estimate roughly how many ways we can do that well if you look at all the ways in which things might be the same they sort of depend on automorphisms of this vector space and this vector space and maybe automorphisms of the whole lot and the automorphism of vector space forms a group gl of m prime and its number of elements is about p to the m prime squared that's the number of matrices of size m prime by m prime each of whose elements isn't the finite field of order p and this is going to be at most quadratic in n so we should really take p to the two 27 n cubed minus something quadratic in n so to account for the the fact that we've constructed many of these groups several times well this quadratic term in n is maybe quite large for n small but it's eventually going to be dominated by this so what we see is the number of groups of order p to the n grows like p to the two over 27 n cubed i don't mean it's asymptotic to this because this this term here means it's actually less than that but this is the sort of main main term driving the growth is roughly this size in fact we see the number of groups of order two to the 10 that we had doesn't really even that doesn't give us an idea of just how vast this number is because if we take p to the n equals two to the 10 we find this number here is about two times 10 to 22 which is much bigger than the number of groups that actually are of order two to the 10 so it's saying see that even for groups of order two to the 10 this quadratic term here is still keeping this cubic term under control but as n gets larger and larger this cubic term starts to dominate and the number of groups grows even more rapidly than um and this number here suggests so there's just no great incentive to classify groups of order p to the n for n large there are just too many of them okay well i'm going to stop going through orders of groups because from after about 32 um the not that the orders where something interesting happens for groups of that order become increasingly rare and what i'm going to do is just jump to the next most interesting group so next lecture we're going to be looking at groups of order 48 and groups of order 60