 This lecture is part of an online mathematics course on group theory and will be about some analogues of the Heisenberg group that are called, rather unimaginatively, extra special groups. I didn't invent this terminology so don't blame me for it. So in previous lectures we've classified groups of order up to 24. So let's have a quick look at groups of order 25, 26, 27, 28, and 29. And most of these nothing new happens. 25 is of order p squared, so there are two Abelian and nothing else. 26 has ordered two times a prime, so we get a cyclic and a dihedral group. 29 is prime, so we only get a cyclic group. 28 you have to think a little bit more about, but this is of the form four times a prime, so it turns out to be very similar to the case of groups of order 20. And in fact we get four groups, two Abelian ones, one dihedral one, and one binary dihedral one. So order 27 is the only order that might give some new phenomena. So while 27 is three cubed, so what we're going to look at is groups of order p cubed for p prime. And first of all there are three Abelian ones, which are not terribly exciting because we have z over p cubed z, z over p squared z times z over pz, and z over pz all cubed. So what we're going to discuss are the non-Abelian ones, and it turns out there are always two non-Abelian ones. If the group is non-Abelian, there are two possibilities, that it can have elements of order p squared, or all elements have order one or p, that's because the order must divide p cubed, and it can't be p cubed because then the group would be Abelian. Now if p equals two, we've already classified them, and there are two dihedral group of order eight, and the quaternion group of order eight, and there are no groups such that all elements have ordered two because we saw that if all elements have ordered two, the group has to be Abelian. If p is odd, then there is one group with an element of order p squared and one group of elements of order p. So although for both p equals two and p odd, we get two non-Abelian groups, they don't really quite correspond. The classification is different for p two and p odd. So in any case, we can make some easy observations about the structure of the group. If g is a group of order p cubed and is non-Abelian, the center of g is of order p, and must be isomorphic to z over pz. That's because any p group with more than one element has a non-trivial center, and the center of a non-Abelian group can't have index a prime, as we saw earlier. So it must have index p squared and must be this, and let's call the center z, and then g modulo z must be isomorphic to z modulo pz squared. I'm using z in two different ways. Here it means the integers, and here it means the center of g. Sorry about that. It's standard terminology in both cases, and that's because g modulo its center can't be cyclic unless it's trivial. So g looks, sits in an extension like this. One goes to z modulo pz, goes to g, goes to z modulo pz, all squared goes to one. So it's an extension of an elementary Abelian group of order p squared by a cyclic group of order p, and we have to figure out what it is. First, let's look at the case when all elements of order one or p. First of all, are there any examples of such groups? There aren't for p equals two, but there are for p greater than two because we can take all the following matrices that are three by three matrices. So this is in gl three of fp. This is the finite field with p elements, and this is obviously a group of order p cubed, and you can easily check it's non-Abelian. And the question is, do all its elements of order p? And if so, why do they stop having order p when p is equal to two? Well, if you look at this group here, sorry, this vector space, so here we've got a vector space, and we can define an exponential map from this to our group, and the exponential map is just given by x of A is equal to one plus A plus A squared over two factorial, and we notice that A cubed is equal to zero if A is one of these matrices. So the exponential map just looks like that, and it's got an inverse, which is the logarithm map, where log of one plus A is just A minus A squared over two, and we notice that A cubed is equal to zero. So the exponential and logarithm maps are just polynomials. And x of A plus B is equal to x with A times x with B if A B equals B A. In general, this doesn't hold, but it's at least true if A and B commute. So in particular, x of NA is equal to x of A to the power of N. And now, if we've got an element g in this group here, let's work out what g to the p is. Well, this is equal to x of p times log of g, which is equal to one because p times log of g is equal to zero because p times anything is zero since we're working over the field with p elements. So we see that all elements of this group here have order p, except when p equals two, because we need to divide by two factorials. So this only works if p is not equal to two, and this only works if p is not equal to two. So we've got a non-Abelian group of exponent p, unless p equals two. What happens if p equals two? The group just becomes the dihedral group of order eight and does indeed have elements of order four. So this argument breaks down. So this, remember when we did groups of order four or possibly eight, I think we commented that if all elements of a group of order one or two, then the group is Abelian. And we now see this fails for elements all having order one or three. So we've got a group whose elements all have order one or three, and it's not Abelian. We can easily check that there's only one such group of elements of order p, because any group of order p cubed, let's take it non-Abelian, and g to the p equals one for all p, you can see it must actually be a semi-direct product z times pz times z modulo pz, so that's z, with z modulo pz acting on this group here. Well, this is a two-dimensional vector space over a field of p elements, and its automorphisms are just two by two matrices, and the automorphisms of order p are not too difficult to classify, they're all conjugate to this automorphism here. So up to isomorphism, this element, we may as well take a generator of this element to be acting on this group here by this automorphism. So we see that this group can be given as follows. It has elements a to the p equals one, b to the p equals one, so a and b are generators of this, a, b equals b, a, so they commute with each other, and then we take an element c here, and we know c to the p equals one, and c commutes with a, cb is equal to a times bc, so you can now check that these generators do in fact define a group of order p cubed fairly easily, so this shows there's most one group of order p cubed, that's non-abelian, has all elements of order p, unless we found one for p odd, it must be that group. So there's a unique group of order p cubed for p odd with these two properties. Now let's move on to the groups where g is order p cubed, and is non-abelian, and has an element of order p squared, and let's see if we can find all such groups. Well g has a normal subgroup z modulo p squared z, so this is contained in g, so g is generated by, so z modulo p squared is generated by an element a with a to the p squared equals one, and now let's pick b not in the c modulo p squared z, and then what we know about b and a, well b has to act non-trivially on a, otherwise the group would be abelian, so bab to the minus one is equal to a to the something, and the automorphisms of z to the p squared z are all of the form a to the one plus something times p, and we can simply replace b by a power of p, and we may as well assume that it takes a to a to the one plus p, if it takes a to the one plus two p or three p, we simply change b to something else so that it's like that, and then what is b to the power of p? Well b to the power of p must be an element of order one or p here, otherwise b would have ordered p squared, so it must be one a to the p a to the p squared and so on, and you can see that the group is determined by these relations, so the only ambiguity is what is b to the power of p? Well if b to the power of p is one, then g is just a semi-direct product, z modulo p squared z times z, semi-direct product, sorry z, quotiented out by p z, and if p equals two then this case here really does give us a different group, so p equals two, this gives us the quaternion group, this is the dihedral group if p equals two, so we really get two, we really do get two different cases of p equals two, it looks at first sight as if we're also getting several different cases if p is odd, but it turns out that if p is odd then the group with b to the p equals a squared turns out to be the same as this group here, the point is we can change b to, let's call it c equals b times a, and this element has similar properties, you can see that cac to the minus one is still a to the one plus p, so the question is what is c to the power of p? So let's calculate it, well we know that bab to the minus one is equal to a to the one plus p, so c to the p is ba to the p, that's ba to the p which is ba, ba, ba, and so on, now a to the p commutes with everything, so what we can do is we can swap around this b and the a using the fact that b to the a equals ab times a to the p, so every time we swap around an a and a b we pick up an extra factor of a to the power of p, so this is equal to a to the p times ab, a, ba, ba, and so on, and then we get a to the p squared a, b, a, b, ba, and then we get a, sorry, not a to the p squared, it's a to the 2p, then we get a to the 3p times a, a, b, b, b, a, and so on, so we keep moving b's to the right, and every time we swap around a b and an a, we pick up a factor of a to the power of p, so we continue like this, and we get a to the p times p plus 1 over 2 times p times a to the p, b to the p, that's because we have to exchange a's and b's, p times p plus 1 over 2 times in order to get all the b's to the right, now if p is not equal to 2, this is of the form a to the p squared to the p plus 1 over 2, which is 1, so this is just equal to a to the p, b to the p, so so we had these relations b to the p is a to the p, but c to the p is going to be b to the p multiplied by a to the p, so this changes the p's power of our element by a factor of a to the p, so if b to the p is some power of a, so if it's a to the np say, then by changing b to b times a, we can get an element c to the p which is a to the n plus 1 times p, and by doing this repeatedly, we can just get an element, a new generator say d such that d to the p is equal to 1, so our group has relations a to the p squared equals 1, d a d to the minus 1 equals a to the 1 plus p, d to the p equals 1, so our group is a semi direct product, so we really only get two groups of order non-Abelian groups of order p cubed, and we can sort of picture them as follows, so we have two obvious ways of constructing a group of order p cubed, we can take these matrices and the second way is to take a semi direct product, p squared z, semi direct product z over p z, and we can look at these for p equals 2, 3, 5, 7, 11 and so on, and for this sort of group we get one example, and we can sort of think of these as being a family, and we also get a group like that for p equals 2, and we get a group like this for p odd, however for p even these two groups are actually isomorphic, and we get a new group which is the quaternion group q8, so this one, this group here is d8, so this is a sort of picture of the classification of groups of order p cubed, we get 2 for each prime, but we really get two families which are the different from most odd primes and happen to be the same for p equals 2, and we get a third family which only works for p equals 2, well these groups are the analogs of the Heisenberg group in quantum mechanics, so let's recall what the Heisenberg group is, I suppose you look at all functions, so f is a function from the reals to the complex numbers, and then you can transform these functions in two ways, you can take f of x to f of x plus a, so we just translate the function left or right, and we can also change the function f of x to e to the 2 pi b x times f of x, so we just multiply it by this periodic function, and in quantum mechanics one of these, I think this one corresponds to a momentum operator, and this one corresponds to a position operator, or maybe they're the other way around, I can't quite remember, so if you call this transformation ta, we call this transformation tb, we see that ta and tb don't quite commute, we can do them in one order, or we can do them in the other order, and these differ by a factor of e to the 2 pi a b, which is just multiplication by a complex number of absolute value one, so that should be tb times ta, so we get a three-dimensional group of all operations where we take f of x to e to the 2 pi i c plus b x times f of x plus a, and this gives us a three-dimensional group of transformations, and you can see that it's got a subgroup isomorphic to the circle group, just a multiplication by numbers e to the 2 pi i b x, and then it sits like that, and the quotient is just a product of two copies of the reals, because these two translations commute up to multiplication like that, so that's the Heisenberg group in quantum mechanics, it's an extension of r times r by s1. Now let's do this over a finite field, so f is now a function from a finite field to the complex numbers, and we've got these translations ta which take f of x to x plus a, where a is now in the finite field of order p, and we've got tb takes f of x to e to the 2 pi i b x over p times f of x, and this is b times x is only a number modulo p, but e to the 2 pi i has period one, so this number is actually a well-defined p through to unity, that's why we put the 2 pi i in up there, and these commute with each other, opt to multiplication by something of the form e to the 2 pi i c over p for c in the finite field, and this is well-defined because e to the 2 pi i x is periodic, so we get a group of order p cubed, which sits like this, so we just replace r by fp and replace s1 by fp, and this is now just one of the groups of order p cubed, in fact it's usually the one, it's isomorphic to the one of upper triangular matrices, so the point is that groups of order p cubed are very similar to the Heisenberg group, for instance if you look at their representation theory, the representation theory is very similar, well Heisenberg groups exist not only in one dimensional, I mean you can also do Heisenberg groups in n dimensions, and these are rather similar, except it's an extension, oops I missed out the g, one goes to s1 goes to g goes to r to the n times r to the n, so you can do the same thing except you replace one dimensional space by n dimensional space and you get a group like that, and you can do the same thing over finite fields, you get a group fp goes to g goes to fp to the n times fp to the n goes to one, and groups similar to this are called extra special groups, so what's an extra special group g, well g has order p to the one plus two n, the center is order p, let's call the center z, and g modulo z is isomorphic to z over pz to the two n, so these are the analogs of the Heisenberg groups, they should have been called Heisenberg groups, not extra special groups, but we're stuck with this rather rather silly name, so what can you do with extra special groups, well they they turn out to be quite easy to classify which is quite convenient because they they turn up quite often in groups theory, so I'll just sort of sketch how you classify them, it turns out you can build them all out of groups of order p, first of all if we take g over z which is z modulo pz to the two n, this actually has a skew symmetric form which takes ab to ab a to the minus one, b to the minus one, and this is now in the center which you can think of fp, and you can check that ab is equal to ba inverse or minus ba if you think of this as being in the finite field like that, so what we've got is really a vector space of dimension two n over a field with the skew symmetric form, and you know how to classify skew symmetric forms, skew symmetric forms are just a sum of forms with the matrix nought one minus one nought, so any skew symmetric form on a on a field if it's non-degenerate can be split up into a sum of two-dimensional forms, and similarly g sort of splits up as a central product of groups of order p cubed which we've classified there are just two of them, so a central product of two groups a and b means you take the product a and b and you identify the centers of a and b, so we're quotient out by things of the form z1 minus, so z1 prime to the inverse where we're given an isomorphism between the center of a and the center of b, well groups of all p cubed all of center equal to z so you can identify the centers and so by taking a product and taking a suitable quotient of the center you can construct all extra special groups from the groups of order p cubed, it turns out we get two extra special groups of order p to the one plus two n and these can be distinguished as follows for p odd one has all elements of order p and one has an element of order p squared, so for p even we again get two groups but you can't distinguish them like this because one always has order p squared and you can distinguish them using the aft invariant, so I'll finish this lecture by just telling you what the aft invariant is, so if you've got one of these extra special groups of order two to the one plus two n so g over z is a group z modulo two z two n and this has a quadratic form q on it, so we've got a map z modulo two z to the two n goes to z modulo two z where this is the center and this quadratic form is defined very easily q of a is just equal to a squared you know that the square of an element in g will be an element in the center and we can check that q of a plus b is sorry q of a b is q of a q of b times a b where this is the skew symmetric product a b a to the minus one b to the minus one so we've really got a quadratic form on a vector space over a field with two elements and quadratic forms on f two to the n to f can be distinguished by the aft invariant, the aft invariant is nought or one and it's given by the most common value of q it's sometimes called the democratic invariant because you just take a vote of all the values it takes and that's the aft invariant so for instance in q eight the quaternion group if you look at g modulo z that's a group of order four and the values of the squares of these elements nought one one one you remember that i squared j squared and k squared are all none zero so the aft invariant is one because one occurs most often if you look at the dihedral group and work out the squares of these elements um sorry the squares of the elements of d eight modulo the center three of them have ordered two so their square is the identity element and one has order one so the aft invariant is zero so the aft invariant the quaternion group is one and the aft invariant of the dihedral group is zero and similarly for all other extra special groups of order two to the n the aft invariant will distinguish them okay that's enough about extra special groups next lecture will be about a group theoretic map called the transfer which we will use to classify the groups of order 30