 So you might ask what is one of the first things that flow from this one of the first things is a law Let's call it the cancellation cancellation law This is have a quick look at the cancellation law or we're gonna have is something like this. We're gonna say a b equals a C that implies That b equals c. What is that b equals c? You know what that was but anyway It's easy from all those properties that we set to be able for something to be able to be a group Because I can multiply both sides or I can bring this operation and this operation is going to be This operation of multiplication so I can bring a inverse in on this side and I still have a b and On this side I'm gonna also bring in a inverse and I have a c look at the order of that This comes in from the left-hand side now. We had this property of a associativity So I've got a inverse a and then b and that is going to be equal a inverse a And c and we had that property that we must have the identity element in there so eb equals ec and we had this property of there must be this identity element and The product of that is just going to be b and that is going to equal on the other side c So that is the cancellation law and that follows Naturally one of the first things that then we do follow That does follow from that. Can we use the cancellation law? Ah, let's use it in a nice fun little Exercise here. Let's suggest that if a b equals the identity element this implies That a equals b inverse and b equals a inverse Can we prove that from the stuff in our bucket the properties that we've had? Let's let's have a look proof is Given what is given well given that a b equals e What can we do to this? We can multiply this side by a inverse a b and then equals a inverse e Associativity we can bring into this So eb Anything with identity element is this leave that for me. This is just this b equals a inverse So I'm taking a group here in my group operation. By the way, it's this multiplication allowing me to do this Okay, remember though that this is not it doesn't really mean the algebraic form that we used to that you used to from simple Algebra it says it is just a notation for the inverse remember this was addition and under the integers and 3 and Was my element in minus 3 would be its inverse But I could also just have said this was a this was my a inverse not necessarily saying this This makes sense for us algebraically what we mean here is I'm just Generally implying what the group operation is here, but I'm generally generally talking about an inverse Now we've proven one, but we haven't proven the other implication here and We can really simply do that. Let's see if we can do that. So that's proven on that side now. Let's also given Given what is given to us given that a b equals e. I can have this a b and let's have Be inverse on this side So I have this and I have e and I have b inverse on that side Associativity which is this going to be a and you can clearly see here We're going to be left with e and this is a and on this side. We can have b inverse So with the cancellation law and those properties This is what it implies if I take two elements of my group with a group operation Applying the group operation to those if that equals the identity element It means the one is one the one element is the the one element is The inverse of the other