 Ok, ek denk dat ek nu genoemd is, ek heb genoemd is in ons bucket dat we in groeps gaan kijken. Eventuurlijk gaan we daar nie genoemd is, we nie genoemd nie genoemd is in ons bucket. We gaan wat meer, we gaan revies wat in het bucket is, maar het bucket is vol genoemd is nu dat we in groeps gaan kijken. By the way, griembord. Whiteboard gond. Now, the whiteboard, ek mean the whiteboard looks nice but it's, I find it awful to write on. I want this, I want this connection to something and I get that connection by this resistance between the chalk and the chalk. I just love a chalkboard. Anyway, let's get to our definition of a group. What is a group? First of all, it is a set of elements. So I've got to have a few elements and we won't go into depth now about Zermelo, Frankel set theory. This is, imagine we have this set or this set of elements. So it's, let's just call it the set of elements, set of elements and the operation that we can apply between elements of that set. Now in order for this set and its operation, so imagine the integers under addition or the integers under multiplication, set of elements and the operation. I have to have the following property. First of all, there's got to be closure. In other words, if I take any two elements in that set, so imagine my group is the set g and I take two elements, let's make it g1 and this operation and g, let's make it gi and g sub j, so that's two elements in this group and I apply the operation that this, whatever I get from this operation must be an element of g. So the solution or the answer of taking two elements and this group, this operation on these elements, that solution must also be in g, so there's closure. The second one is really is there must be associativity or sociativity. In other words, if I have this scenario where I have g, let's make it g1 and this operation g2 and the operation on g3, just three random elements from my group that I have the following, that if I do these two first and then that, that has got to equal g1 and then g2 and g3. If I have that, I have associativity, that is different from commutivity. You can have commutivity, you can have groups that have commutivity, but it's not part of the definition here of a group. Number three is one of the elements must be the identity element, must be the identity element. So one of these elements in g must be the identity element, so that if I take any random element in g and I have this element, I have this one of this elements must be the identity element and that gives me g or e, g gives me g. I must have this property that one of these elements must be the identity element and then lastly we must have the inverse element, inverse element. So there's one of them must also be the inverse element. In other words, if I have the inverse of g1 so that I have g1 and my operation and it's inverse that I must get the identity element and the same has got to go for the inverse of that and g1 that's also going to give me the identity element. So that wasn't that difficult, was it? If I have all of these things, I have the set of elements with an operation and it has these properties, then I have a group. I have a group and we can think of a group. Let's just think about the integers under addition, the integers under addition. Of course I have closure, in other words any two integers that I add to each other, the result will still be in there. I have associativity and as much as 1 plus 2 is 3 plus 3 is 6 and 2 plus 3 is 5 plus 1 is 6, exactly the same thing. I have this identity element which in this case is going to be zero and for any element in that group a I have the inverse which is just the negative of a. So these integers under addition for me for us would be a group and it seems so trivial but it's actually so powerful.