 Great stuff. Let's do some exercises just to remind ourselves that we are not dealing here with algebra You might be asked just to solve for x there I have a x and b equals c and then for all a b c x elements of this group And I'm gonna write group like this with these little brackets the group So that's all the elements all the elements in the CG and the operation on those now I just want to remind you I mean if we look at a x b we could say well We could just say in normal algebra. I was just going to say that c equals a C equals x equals c is as a over b or x equals. I mean I could write this as c or You know bring it up the other side so that c a inverse b inverse But that would just suggest that these are numbers and they knock this is not representative the elements here are not representative of numbers if these were for instance square matrices I want to have that we do not have this commutative property. So remember we have closure So if this group operation if I apply that to any two elements in the set the solution must also be there I have a sociativity a sociativity. I have identity Identity element and then for every element. I have its inverse inverse also in that set No way that doesn't say commutative. So just be aware when you do this So what I'm going to do here is say a inverse a x b equals a inverse c a inverse means my associative property that this gives me the identity element so that disappears I can have x b and b inverse meaning I have a inverse c b inverse That a sociativity that gives me is the identity element. So I have x equals a inverse c b inverse And if these were anything these can now be any elements for instance square matrices if I did them in this order I'm going to get a solution to x otherwise I would not have gotten a solution to x if I just saw this as normal algebra. So please please please remember those Let's do some more so we can so we can get used to to this idea So let's have a look at this problem now try and get some of these online or in your textbook They really are fun to do just to get your mind around working with groups when I write x squared They remember I have my group and I have Multiplication is my operation here. So when I have x squared I actually mean x and x So I have this fact that if I do that the group operation on two of the same element They give me b and if I do it five times I get e in other words What I'm suggesting is x and x equals b and What I have here is x times x times x times x times x equals the identity element e So for two of those x's I could put a b for two of those x's I could put a b and there I have an x and that equals e So I have this b I suppose I could do this in two ways I could say b inverse b x equals b inverse e so that goes away. I can do it again. This b x equals now I have b inverse another b b inverse there from this free b inverse And I have the fact that I now have this b inverse Squared and that's actually just this b squared. It's inverse But just to remind yourself that we are not dealing with just Placeholders these are not mathematical variables and they are not placeholders just four numbers. Let's have a look at another one Now let's have a look then at this one. I have x squared a equals b x c inverse and a c x equals x a c I've got to solve them simultaneously not through normal algebra This is see what we can do here Let's see I can do a b x there You just have to look at these and start playing quite a bit of fun to do So let's play. Let's put a c there. So I'm gonna have x squared a c Equals b x because if I put a c at there c inverse c that gives me the identity element And what I could do here is I could have an x I could have an a c inverse So I put an a c inverse in the front put an a c inverse in the front or sociativity x a c I can put a b there and I can put a b there So I have b x and I have b x at least we have that so we have at the moment x squared a c and B x and b x that's the same. So that's going to be b a c inverse x a c Immediately I see I have the a c here and a c here So I could do this a c associativity and a c inverse which is just going to give me an e And if I do exact same thing there, of course, I'm just going to be left with a c inverse x on that side So what I have here essentially is x and an x and that equals b Let me make some more space because you can clearly see where this is going that it's going to be b a c inverse x And what I can do here is an x inverse on both sides x inverse on both sides And that's identity element identity elements. I'm left with x equals b and I have a c inverse Dunn-dusted. These are really a lot of fun Really look them up as I said try to get some online in your textbook and try to do some of them Just to remind yourself that this is not simple algebra anymore We are talking about a group set of elements and then the the operation on those elements And there's a group. So we are dealing with those properties