 So this video is part of a mathematical online course on group theory and this lecture will be on homomorphisms of groups. So we will just recall that a homomorphism f from a group g to h is just a map from g to h that preserves multiplication. So f of ab is f of a times f of b. A consequence of this is that f of 1 is equal to 1 where this is the identity of g and this is the identity of h and f of a to minus 1 is the inverse of f of a. So you can think of a homomorphism as being a function that preserves the group structure. It's called an isomorphism if f is by a bijection. This means that f is onto and each element of h is the inverse of a unique element of g. So in particular there is an inverse function f to the minus 1 from h to g. It's called an automorphism if it's an isomorphism from g to itself. And finally the kernel of f is the set of elements a with ga equals 1. So an isomorphism between two groups g and h sort of means that g and h are in some sense the same group. I mean they're the same group except the elements have been re-labelled. So an automorphism of a group can be thought of as a symmetry of the group. So you remember a symmetry is just a bijection from some mathematical object to itself that preserves whatever structure. So these are just symmetries of groups. In particular a set of automorphisms of a group is itself a group because it's the set of symmetries of something. So having reviewed those basic definitions let's just review the standard examples. And the first example that most people come across is just the exponential map where x with x is as usual 1 plus x plus x squared over 2 and so on. And this has the property of x plus y is x of x times x of y. And here we have the group operation of the real numbers under addition and this is the group operation of the group of non-zero real numbers. So this means non-zero reals under multiplication. So when I wrote the definition of a homomorphism I said that f of a b is f of a f of b. A b should sometimes be replaced by the group operation if you're not writing it as multiplication. So here the group operation is written as addition so we change the definition of homomorphism slightly. So it's not a bijection because it's not onto. The image of x is always positive. However it is an isomorphism from the real numbers under addition to the positive real numbers. So let's call this positive reals under multiplication. So the positive reals under multiplication form a group and this is essentially the same as the group of reals under addition because we have this exponential map that is an isomorphism. So here we've got this exponential map and of course everybody knows the inverse of the exponential map is called the logarithm or ln if you want to use this dreadful notation. So the next example comes from linear algebra and we recall we have a determinant of a matrix. So for a two by two matrix this is just going to be ad minus bc and there's a similar formula for larger matrices and the basic formula is that the determinant of a product of two matrices is the product of their determinants. This means the determinant map is a homomorphism from the general linear group of dimension n over some field may as well be the real numbers to the nonzero real numbers. The real numbers can be replaced by any field. So you recall this is the invertible linear maps from rn to rn or at least it can be identified with a group of invertible linear maps. So there's a homomorphism from this group on to the nonzero real numbers. It's on to at least if n is greater than one if if n is greater than or equal to one. If n is zero then this is just a trivial group and this map isn't on to which is a stupid special case everybody forgets about. The kernel is called sln of r the special linear group which is the special linear group is just defined to be the set of matrices of determinant one. Special by the way in linear algebra often means determinant one so the special orthogonal group is the set of orthogonal matrices of determinant one and so on. So the next example will come from number theory. So we recall that we have this group z over four z which is the integers modulo four. So it's got four elements nought one two three and the group operation is denoted by addition and you subtract multiples of four whenever you add two things. So this is a little little group of order four. On the other hand we have a group z over five z star so this is the nonzero integers mod five under multiplication. So it's got four elements one two three and four and the group operation is good by multiplying them and if the result is bigger than five you reduce modulo five. So you can recall from any number theory course that these are both groups and these two groups are in fact isomorphic and you can have an isomorphism which takes zero to one one two two and two four and three to three. So this is going to be the definition of the map f and the point is that f of a plus b is equal to f of a times f of b so again we're changing addition to multiplication and in fact f of a is just going to be two to the power of a so you can see this is two to the one this is two squared and this is two cubed at least if you're working modulo five um so this is a sort of exponential map to um except where the base is two rather than e um one thing people sometimes do with groups is write out a multiplication table so if you write out the multiplication table of the first group under addition it looks like this so this means that two plus one is three and so on and we can write out the multiplication of the second one it looks like this um so um we get two four one three three one four two four three two one and if you look at these you don't get the second table from the first by by changing zero one two and three to these numbers here so um a common mistake people make when they're starting over the group theory is they these two multiplication tables um are not the same if you change zero one two three to one two three four you can see that things go wrong here so these two groups are not isomorphic that's a mistake because you can also switch the order of rows of a multiplication table so if I switch these two rows and these two columns then the two multiplication tables really do become the same if you just relabel these elements according to this rule here so um you can't it's difficult to tell whether or not two groups are isomorphic just by staring at their multiplication tables in fact writing out the multiplication table of the group is nearly always a rather stupid thing to do it gives very little information and for any group of order larger than about four it becomes ridiculously tiresome to write out the multiplication table so my advice for dealing with group multiplication tables is just don't bother with them there are a lot of work and don't really give you very much useful information um the elements of z modulo four z are automorphisms of um z modulo five z here we are identifying z modulo four z with z modulo five z um the non-zero elements of this um for example if you take um an element of z modulo five z and map it to two times g this is an automorphism of z modulo five z the reason being that it has an inverse g goes to three g because two times three g is six g which is equal to g in z modulo five z so this gives several examples of automorphisms of groups another example of a homomorphism is um the a map from the reals to the circle group so the circle group s one its elements are just the elements of the unit circle so that pairs x y with x squared plus y squared equals one and if you fix a point you can think of an element of this group as corresponding to a rotation of the plane by this angle so the points of the circle correspond to possible rotations of the plane fixing the origin and the group multiplication is easy enough to work out if you've got two points x one y one and x two y two on the circle then the product is going to be x one x two minus y one y two x one y two plus x two y one you can see this in two ways you can either think of x y as being the point cosine theta sine theta and then um if x one y one is cosine theta one sine theta one then this becomes cosine theta one cosine theta two minus sine theta one sine theta two which is the usual formula for cosine theta one plus theta two and similarly this becomes sine of theta one plus theta two which i'm not going to write out explicitly so um we have um a nice group operation on these numbers here and there's a homomorphism from the real numbers two s one which just takes a point um theta of r to cosine theta sine of theta and um we've um the um group operation that we've defined here just makes this into a homomorphism by the formula for cosine and sine of an angle so um if we call this f then f theta one plus theta two is equal to f of theta one f of theta two where this is the product given above um next we can ask what is the kernel of this homomorphism well the kernel consists of all real numbers theta such that cosine of theta equals one and sine of theta equals naught so it's just multiples of two pi that's integer multiples of two pi um for a final example of a homomorphism um let's look at a slightly less obvious one what we're going to do is to take the group of rotations of an octahedron so let's take the rotations of an octahedron which has order 24 meaning it's got 24 elements in it and we're going to have a homomorphism from this to the group s3 which is permutations of three points and this is order six um and what we're going to do is we're going to take three objects what I'm going to do is I'm going to take the three diagonals of an octahedron so you see it's got one diagonal between my finger and thumb here and it's got another diagonal here and a third diagonal here and here so these three points are going to be the three diagonals and any rotation of the octahedron will just permute these three diagonals so this homomorphism in some sense trivial um any rotation of an octahedron automatically gives us a permutation of the three diagonals and if you think about this for a few seconds you see this is just a homomorphism and what's the kernel of this homomorphism well this group here is order six and this group is order 24 so there are definitely some elements of this mapping to the same element of this and we'll see fairly shortly that the kernel of a one group mapping on to another has ordered the quotient of this number by that number so the kernel should have four elements and we can see these four elements here's one of them I picked this diagonal and I just wrote by 180 degrees about this diagonal and you can see it maps each diagonal to the to the same diagonal and there are three ways of doing this because there are three diagonal so the kernel is the three rotations by 180 degrees about the diagonal plus the trivial symmetry where you do absolutely nothing to the octahedron so that's enough examples of homomorphisms the next few lectures what we're going to do is describe the small finite groups in order of size making a sort of rather forlorn attempt to classify all finite groups and what we'll do is we'll introduce lots of theorems about finite groups whenever we need them to classify finite groups of some order