 So let's look at an example then of an isomorphism and an isomorphism is a bijective homomorphism and you'd notice that we had some colour chalk today, today we are very very fancy so let's have a look I've got my two sets here, G is this integers mod 4 under addition and H is my integers mod 5 minus this group minus this congruent set here 0 so 0 is excluded so instead of writing each one of these I'm just writing it as that with the line under just to make it easier my mapping is such so here's my elements in G it is the integers mod 4 so that's there, the integers mod 5 1, 2, 3, 4 because I've taken away this congruence class 0 so 0 maps to 1, 1 maps to 3, 2 maps to 4 and 3 maps to 2 is that an isomorphism well first of all, you know, if it is a bijection it's got to be 1 to 1 and onto and one good way to remember this here is 1 to 1 1, 1 to 1 and here's onto remember 1 to 1 is an injective and onto a surjective colored chalk so we have this sort of thing let's make it like that so here's my domain and I map to but not everything here is being mapped to whether onto means more than 1 can map to the same element of my second set and then if it is both 1 to 1 and onto there's a direct 1 to, you know, direct relationship 1, 4, 1, 1, 4, 1 and it's onto so it's both of there so if it is going to be an isomorphism the first thing we better just notice is that the size of the two sets are exactly the same so I have four elements in this set and I have four elements in this set if I start with something that's not like that it's not going to be an isomorphism as far as these examples are concerned that is what we need so first, I think Justin is an example just to remind ourselves, let's have Kayleigh's table with this and we're going to have 1 to 1 0, 1, 2 and 3 0, 1, 2 and 3 and remember these are all my congruence classes so it should actually be that I'm going to leave out even the underlinger this is my identity element there so that's going to be the same 1 plus 1 is 2, 2 plus 1 is 3 3 plus 1 is 4, mod 4 is 0 and then 1 and 2 is 3 2 and 2 is 4, mod 4 is 0 it's 5, 5 mod 4 is 1 and then 3 and 1 is 4, mod 4 is 0 it's 5, mod 4 is 1 it's 6, mod 4 is 2 let's do the multiplication so this was G let's do H very quickly and we're going to have 1, 2, 3 and 4 there 1, 2, 3 and 4 there and 1 is for multiplication 1, 2, 3, 4 it's my identity 1, 2, 3, 4 there let's do which one? 2 times 1, 2 times 2 is 4 2 times 3 is 6, mod 5 is 1 2 times 4, mod 5 is 3 and then we have 3 times 1 is 3 3 times 2 is 6, mod 5 is 1 3 times 3 is 9, mod 5 is 4 3 times 4 is 12, mod 5 is 2 and 4 and 1 is 4, 4 and 2 is 8 mod 5 is 3 4 is 12, mod 5 is 2 16 mod 5 is just a single 1 so let's just check this if we have an isomorphism so what I'm saying here is let's just take any 2 let's make it 2 plus 3 so if I have 2 plus 3 because that's under addition I get so 2 plus 3 is 5 mod 4 is 1 so that's going to give me a 1 let's see if this is the same if I do this now under H so 2 maps to 4 that's multiplication for this one 3 maps to 2 and what is 4 times 2 so 4, binary 2 that gives us 3 3 and so 1 better map to 3 and indeed 1 maps to 3 and you can go through all of these and you'll see that all of them holds and so here we have a beautiful example of an isomorphism because it's a 1 to 1 homomorphism here for G and H and try and remember these what is 1 to 1 onto that always helps and just always some practice in making these especially when it's integers mod n so here we have a beautiful example of an isomorphism now for what lays ahead I want you to notice one extra little thing have you ever noticed that if I look down any row if I look down any row of these groups every element in my set appears and all of them appear so I have 0, 1, 2, 3 I have 0, 1, 2, 3 I have 0, 1, 2, 3 0, 1, 2, 3 0, 1, 2, 3 0, 1, 2, 3 if you ever notice that that's a very important that's a very deep thing that's happening here and it is used in proofs for instance when we are eventually going to get to I'll see when we get to that Cayley's theorem Cayley's theorem here's our Cayley's table Cayley's theorem and we're going to use use that sort of thing but just notice that it will also help you when you construct these Cayley's tables that you don't get these if that is indeed a group that you don't get repeats and all of them are there so with a finite group of course you know if each one is there then they are not going to repeat they are going to be unique because if there is to so many and there's no repeats then they're all going to be there or they're all there there is not going to be any repeats because it's finite but that is something that you can always check on but it has a much deeper meaning here and when you eventually get to Cayley's theorem that would help us a lot