 Now let's just have a look at commutivity. Now remember what I said It's not part of the definition of groups, but you might find that in some groups Some elements or elements to commute. So let's just look at that. So that'll be a special kind of group It's definitely not true for groups in general. So we have this statement We're going to let ab equals ba for all ab elements of some group under multiplication group g under multiplication and we say if this is true If this is true show that the inverses of a and b also commute So the proof for this actually would be just very easy. We are given let's we can write that that's given The fact that ab equals ba And what can we do? Well, let's start bringing things Let's start bringing things this way. So I can put a b inverse on that side. So b inverse It's just ab so b inverse b. That's just e and I have an a I can bring a inverse this side So I have a inverse b inverse ab That equals e Because I put an a inverse there Now let's take the b inverse to that side. So a inverse b inverse a Equals I've got a b inverse there which makes us ease. I've got a b inverse on this side Eventually at the end I'll take an a inverse on both sides. So I have a inverse b inverse equals b inverse a inverse So as simple as that as simple as that. So if we do have this commutative Property of commutivity for this very special group Then the inverses of these two will also commute So in this instance, let's have a look same thing is holding here I'm saying that there is in the special group of mind g. This is this commutivity Just show that ab and a would commute Well, it's given it's definitely given that ab equals ba And I can just put an a on a on both sides at the front. So a and ab equals a and ba and all I have to do now this is a group. So I do have this is this Associativity So I can say that these two and those two and then behold a and ab commute because I have a and ab equals ab and a So that's an easy one Still with this very special group of mine. Let's just show that a squared and b would commute So we're going to say given ab equals ba I can Put a in the front. So I'll have ab ab equals ab a and We know that ab equals ba. So I can say a squared b equals ba a Associativity there ab ab is ba so I can swap those two rounds So I have the fact that a squared b equals ba squared and they commute as well So let's have a look at this one. It's slightly more interesting I want you need to now show that a squared and b squared commute So given that ab equals ba I can say that a squared b Oh ba equals aba It's putting an a in the front. Let's put a b in the back. I have a squared b squared equals ab ab Not exactly what I want, but we can do association ab equals ba So that is going to be ba ba with another a squared b And we've just shown that that b squared and a squared that a squared and b squared commute So those two commute we've just shown that so we're going to have that b squared a squared as I Turn those two around. So I have a squared b squared equals b squared a squared Okay, we've added another special one And we've added x there. So ab and x are element of this very special group and I still have this The property for this special group that we do have this commutivity and I'm asked to do that so given You have to think about this one a bit And of course, you've got to be got to be slightly creative. Let's have ab x inverse and Let's have that And Do exactly the same on this side ab x inverse. So I'm not using a special property here. What is what is there is exactly there? Nothing nothing different. I do know though that ab x inverse that equals x sociativity ba x inverse And now I can just bring in my identity and I know that x x inverse Equals the identity element because this is a group And so let's just bring e in I'm going to bring e in right there in this form. So I'm going to have x It's not square there. I don't know why I have that there and anyway x a let's have x inverse x which is this e b x inverse and that's going to be x and I'm going to bring e in right there So I'm going to have x Let's see. We're going to have x of b x inverse x a x inverse and now I'm just going to use a sociativity There and there so in other words x a x inverse That's one and x b x inverse that is going to be x b x inverse x a x inverse And we see that they commute so just a little bit of creativity And you don't you might just get it the first time except if they if they're quite easy And of course if you're terribly smart But sit around and play with them and it doesn't take too long if you use a bit of Creativity and you play around and these are actually quite a bit of fun to do