 Now that we've seen a couple of groups, very interesting groups, let's let's just add some stuff to our bag a little bit again Our box in which we put stuff and I want to talk about homomorphisms and all the fisms Now we've we've I think we spoke about isomorphism a little bit, but let's put this in a bigger context So I'm going to create this function f and it's going to be a function that maps You know one group to another and if it preserves or it looks at the structural Similarity between the two I call that a homomorphism. So hands up who understood that so let's just look at something I let's construct two groups G and G consists of some set A and under some binary operation and B H is another group B and some totally different Binary operation, so I don't need the same sets I don't need that for A to be equal to B And I certainly don't need for the binary operations to be together, but is there a function that I could Do that will map G to H Such that this function is a homomorphism So it preserves the structure between these two sets if I can if I can create such a function I have a homomorphism homomorphism So let's look at an example of Of some homomorphism. Let's have G and let's have that the set of all real numbers Let's have that a set of all real numbers And I have this under let's make this under Addition and H is going to be the set of real numbers made with positive real numbers on the multiplication So it's not the same set. It's definitely not the same binary operation and I have this F and My F maps every X to e to the power X Every X to e to the power X So let's see if F is indeed a homomorphism on this group's G and H So let's take two arbitrary elements Let's take three arbitrary elements X Y and Z and they are all elements of G And if I write G it means they're an element of my set under that So so what I will have here is a group so there is going to be closure So X plus Y equals Z and my F of X What will F of X then be? Well, F of X is e to the power X F of Y equals e to the power Y and What I'm suggesting here is that I have Multiplication of these two So that I would have this e to the power X times e to the power of Y Where does that come from? e to the power X times e to the power of Y That's going to be e to the power Z That's what I have because my function takes every X and maps it to e to the power X so I better have that and On the left-hand side what I have here is e to the power X plus Y equals e to the power well Z is this X plus Y So those two are indeed equal to each other So this function is a homomorphism between these two Very different groups on this side. I had my multiplication and on this side. I Had my addition so that was under this was under multiplication under this function and the function is this mapping so I think that's a beautiful example illustrating this concept of a homomorphism. Let's clean the board We lastly have put some definitions on the board just to show you this fits into this Isomorphism at least into this broader concept of these homomorphisms So if I have a homomorphism, it does not need to be one-on-one when it is one-on-one One-to-one we call it a monomorphism if it's onto we call it an epimorphism And if it's both we call it an isomorphism So that's where we get our isomorphism and if this function is the mapping of a function of a group to itself And it turns out to be one-to-one we call it an endomorphism And if it's then both one-to-one and on to we call it an Automorphism so if you come across these terms just remember this page It'll tell you what a monomorphism epimorphism isomorphism endomorphism automorphism is based all on this concept of homomorphisms hopefully we get a chance to look at some of the examples at least of this this Monomorphism or at least epimorphisms as far as homomorphisms are concerned