 This is the first lecture of an experimental online course on mathematical group theory. What it's going to cover is roughly what all mathematicians ought to know about discrete groups. The level is going to be roughly somewhere around first year graduate or enthusiastic undergraduate level. The course will be based mainly on examples rather than proofs. I'm going to cover common finite groups and symmetric groups, free groups, reflection groups, and anything else I happen to think of while making the course. Each video should end with a couple of things you can click on to link to the next video and also link to the playlist of all the lectures. So first of all, what is a group? Well, the group is the collection of symmetries of something. So a symmetry of something is just a way of mapping something to itself, preserving all the structure. So we can start with looking at, say, platonic solids. So here we have the dodecahedron, and we can ask what symmetries it has. Well, you can see I can rotate it by one-fifth of a revolution, and it ends up exactly the same as before, at least if you ignore the colours of the faces. So it's got a symmetry consisting of one-fifth of a revolution about this axis, and it's got lots of other axes you can rotate it about. Similarly, you could rotate it about this axis here by one-third of a revolution. So it's got some symmetries of order three, where the order of a symmetry means how many times you have to repeat it to get back to where you started. It's also got some symmetries of order two, because I can take two opposite edges and just flip it about those edges, and it's got a symmetry of order one consisting of doing absolutely nothing. So how many symmetries does it have? Well, if you count, you see it's got 12 faces. So what I can do is I can pick it up and put any one of these 12 faces on the bottom. And once I fix that face on the bottom, there are now five possibilities because I can rotate it up to five times around this axis here. So the total number of symmetries is 12 times five, which is 60. So we say the dodecahedron has a symmetry group of order 60. Well, actually it might have a symmetry group of order 120 rather than 60 because I've only been talking about rotations. And if we allow reflections as well by, you know, picking it up and putting it in four-dimensional space or something, that actually gives an extra 60 symmetries. So we end up all together with 120 symmetries. Similarly, you can work out the number of symmetries of all these other solids. The cube, for example, has six faces each with four sides. So it is 24 symmetries. The octahedron is 24. The icosahedron has 60 and the tetrahedron has 12. If you notice, the cube and the octahedron have the same number of symmetries and the dodecahedron and the icosahedron have the same number of symmetries. That's not a coincidence. We will see fairly soon that, in fact, in some sense, these two have the same symmetry group and these two have the same symmetry group. Well, the simplest example of symmetry groups are just groups of rotations. So if you take something like a star like this, you can see it's highly symmetric and I'm going to sort of put five blobs on it so you can only rotate it and not reflect it. So if you take this star, you can see it's got a symmetry group of order five because you can just rotate it by one-fifth of a revolution like this and there are five things you can do like that because you can rotate it by two-fifths or three-fifths or four-fifths and so on of a revolution. So we say this group is cyclic of order five. And it's called a cyclic group because if you draw a circle and draw points on the circle like that, you can think of you're just rotating the circle by one-fifth of a revolution. Obviously, there's nothing special about the number five. You can have a cyclic group of any given order and these cyclic groups are the simplest possible groups. Another example is the group Z of integers. What's this the group of symmetries of? Well, if you take a line and just draw regular points on it and put arrows on the line and then you ask what symmetries does this object have? Well, the only way to pick it up and put it back down again in the same position is to shift it by a few spaces. For instance, we could shift everything three points to the right and we would call this symmetry three. And similarly for every integer you can shift it that number of points to the right or left. So the symmetries of the line like this give you the integers under addition. Another very famous example of a group is the notorious Rubik's cube. So what you can do is you can have a lot of symmetries. So symmetry consists of twisting it, say something like that or like that or maybe doing that then that. So as is rather well known it has a very large number of symmetries. I'm not going to do any more because I've forgotten how to put it back together except by taking it apart. Another example of a symmetric of a group is the group of all symmetries of a finite collection of points. So suppose I just take six identical objects and ask for what are the symmetries of this set? Well, symmetry means I just move each of these objects to some other object. So one symmetry might be to swap these three objects around like that and maybe I swap these two objects and map that object to itself. So a symmetry is just going to be a bijection from this set of six points to itself. We can ask how many symmetries there are and that's easy to figure out. So the first point we can map to any of these six points. So that gives us a factor of six and once we've done that the second point we can map to any of the five remaining points that we haven't mapped the first point to and similarly the next point there are four choices for it and so on. So we get 720 symmetries. This is called the symmetric group on six points and usually noted by SN or S6 in this case. So another example of a group is the symmetries of vector space. So we just take a field. Let's take the field of real numbers just for simplicity and take a vector space. For instance, we might take r cubed over it and now we can ask for all symmetries of this vector space. Well, what's a symmetry of a vector space? Well, let's say it's just a map from the vector space that preserves addition and multiplication by real numbers and you know all maps from a vector space to itself by linear algebra. These are just going to be given by all three by three matrices. Well, not all of these are symmetries because some of them will map the vector space onto a proper subset of itself. So we want a symmetry to have an inverse and you remember from linear algebra matrix has an inverse if and only if it's determinant is not equal to zero. So the matrices, the three by three real matrices with none zero determinant are the set of all symmetries of r3 and this group is called gl3 of r. So gl stands for general linear group which just means all matrices of none zero determinant. Three means you're working on a three-dimensional vector space and r means you're working over the field of real numbers. So the symmetric groups and general linear groups over fields are particularly important examples of groups in fact you quite often try to understand other groups by relating them to these groups in a subject called representation theory. Another example of group theory comes from Galois theory or complex analysis and I'm just going to give a simple example of this. Suppose you look at the complex numbers and ask what symmetries to the complex numbers have. So we want to map from the complex numbers preserving addition and multiplication and pretty much everything else you can think of. Well, there's one well-known map which is complex conjugation. You just take x plus iy to x minus iy and this preserves pretty much every operation involved in the complex numbers you can think of and this means the complex numbers have a symmetry group of order two. This is called the Galois group of C or more precisely C of the reals because it turns out that in some sense the complex numbers have an enormous number of very weird symmetries that you don't really want to know about. So we really want the symmetries that map the real numbers to themselves. And more generally Galois theory studies symmetries of field extensions where a field extension is just one field containing another field. Physicists are really excited about groups. So in physics we get space-time which is everybody knows is four-dimensional and we can ask what are the what is the group of symmetries of space-time? Well, this turns out to be really important in general rather in special relativity and we get the Lorentz group and the Poincaré group. So the Lorentz group is the group of symmetries of space-time if you fix a point and the Poincaré group is the group of symmetries of space-time if you don't fix a point. So special relativity is really a study of these two groups and there are lots of other groups physicists are rather fond of like the group SU3 which you need if you're studying quantum chromodynamics which is the study of quarks. So SU3 is the set of three by three matrices that are unitary which means if you take the matrix and multiply it by its conjugate transpose this is equal to the identity matrix. So the symmetries of any mathematical object form a group in particular you can even take the symmetries of a group. So suppose you take the cyclic group of order five say the cyclic group of order five for reasons that will appear very soon as denoted by Z modulo five Z and you can represent its elements of zero, one, two, three or four where the group structure is given by adding these numbers and then reducing modulo five and this group has symmetries which is a slightly confusing concept but what we mean is we can map these elements themselves preserving the group structure for instance we can map one to two and one plus one equals two so we then must map two to two plus two which is four we map four to four plus four which turns out to be three if you subtract multiples of five and three maps to one. So this will turn out to be a symmetry of the cyclic group of order five and it turns out you'll see soon that there are altogether four symmetries of this group of order five so groups themselves can have symmetries and this gives you other groups and you can even iterate this if you really want you can take the symmetries of a group and then take the symmetries of that and suddenly get a whole chain of groups or saying that a group is the set of symmetries of something is rather too vague to use for mathematics so we need to write down some axioms for a group so let's think what properties of the set of symmetries of something have well first of all a group is a set with some multiplication and what this multiplication is is just composition of symmetries so if we've got one symmetry and another symmetry we can do the first symmetry then the second symmetry and we get a new symmetry and we're going to call that the multiplication of the group of abc equals a times bc we're going to denote the multiplication of the group by ab it's sometimes known by a plus b or a circle b so if you call it a plus b you shouldn't really call it a multiplication but whatever and it's associative because symmetries are really functions and composition of functions is associative secondly there is an identity element so this is going to be the trivial symmetry and the identity element is known by one or sometimes by zero or sometimes by e and if we denote it by one it has the property that one times a is equal to a times one is equal to a so this is the second axiom and again that's obvious because the identity symmetry is the symmetry that just fixes everything and obviously if you compose that with any other function you just get the function you first thought of thirdly any symmetry of the group has an inverse symmetry which is denoted by a to the minus one or sometimes by minus a if you're writing the group operation additively and a to the minus one just means you undo whatever you did for instance if my group operation turns a fifth of a revolution clockwise I can just undo it by turning a fifth of a revolution anticlockwise so that would be the inverse element and the inverse element is the property that a to the minus one a equals a a to the minus one equals one so these are the three axioms for a group a group has a multiplication it has an identity element and it has an inverse for each element and they have to satisfy these three axioms next we can ask what is the goal of group theory well the ultimate goal first of all classify all groups up to isomorphism well classifying all groups is hopelessly impossible there's just no way we can do it but we can sort of try to classify all interesting ones what does isomorphism mean well sometimes two groups are really the same so when you say two things in mathematics are isomorphic it just means they're essentially the same except you've relabeled them so we'll see some examples later so if you've got two groups G and H you might relabel the elements of G with names of elements of H that just means you've got a function F from G to H and if F is a bijection and if F preserves the group structure so F takes the identity elements of G to the identity elements of H and F takes multiplication in G to multiplication in H and F takes inverses of G to inverses in H then we say F is an isomorphism from H to G that G and H are really the same group except you've just relabeled their elements so we want to classify all groups opto isomorphism we also want to classify all ways groups all way a group is the symmetries of something so this is called representation theory finding all ways a group can act in an interesting way on something is called finding the representations of a group and there are two particularly important cases of this we can either let the group act on a set in which case it's called a permutation representation or we can let the group act as symmetries of some symmetries of a vector space in which case it's called a linear representation so this is the ultimate question how do we find all groups and find all interesting ways they can act on things there's one question that we haven't quite answered we wrote down the axioms for a group and we can ask the question how do we know these are all the axioms are there any other properties of symmetries of an object that we've somehow missed out there's a complete set of axioms for the group in that any set satisfying these three axioms is indeed the symmetries of some mathematical object this is called Cayley's theorem which we will cover in the next lecture