 Now we've seen conjugacy classes and there's actually, we can play with it a bit and a nice way to play with it is just to look at the symmetric group and if we do it the symmetric group there is something that comes out of conjugation that is actually quite important so it's not just all play. So what is the symmetric group and in elements remember that is the quintessential one because that just contains all the permutations of a set so if my set is remember if we had a set and my set was a and a contained one two and three all the permutations of that and I gave each permutation I gave each permutation just a symbol so remember we had identity one we had tau one two we had tau one three we had tau two three we had sigma and we had sigma squared and sigma took one to two two three and three back to one and doing it twice took one to three three two and two back to one so you remember how it's just all the permutations and remember the number that there will be there's three here so the number of elements will just be in factorial so you so so we remember quite easy to do that is that now let's just look at cycle types and what I want to do is use one that's often in the books and that is just to look at s5 so we're going to have five elements and we're going to have five factorial possible permutations and each of them will name something let's just look at a very specific one one that you commonly see in books let's have one two four five and I have one two three four five there and let's just look at one of one of these possible permutations let's take one down to four let's take four goes to three and let's make three go back to one and let's make two go to five and let's make five go to two that's a nice complicated one I hope you can see it back all the way from where the camera is anyway there we go let's do cycle composition of that remember how we do that so the cycle decomposition of it we'll start with the smallest one which is one and we see one goes to four and then four goes to three and three goes all the way back to one so three goes back to one so that's one cycle complete and two goes to five and two goes to five and five goes to two and that is done remember if there is a single one we don't you know if there's just a one we don't change it for identity element we will just write that as one this one we would just write as one two for this s3 remember this is s3 now the symmetric group in three elements this we would write one three we don't write the three there we don't write the two there the reason why is we have these names this is called a this one is called a three cycle because it's just this cycle I can create two cycles here and this one is a two cycle a two cycle and then the one cycles we just we just leave out we just leave them out now let's just consider one other one now I want to show you this one so this will actually have a name this is a three cycle two cycle and that's the cycle type of this permutation now and for for the fun of it we'll just call this sigma that's my sigma permutation so let's just have a look at another one one possible one let's have one two three four five let's have one two three four five and let's do something else let's make one go to two one go to two we'll make two go to three we'll make three go to one we'll make four go to five and we'll make five go to four so we left this one going to two going to three going to one and we left it four going to five note that this is a two cycle and that three cycle I should say and that is a two cycle and look at these two it's a three cycle two cycle three cycle two cycle that is what this one is it's a three cycle two cycle that one's a three cycle two cycle that's its cycle type of this permutation of that permutation and both of them are the same cycle types and there's very something very specific about these two cycle types because what I can show you is I can change this permutation into that permutation just by rearranging this three cycle rearranging this two cycle and it is because it is that cycle type three and two and three and two that I can do this if it was a four is this the four because we won't say four and one but if it was a four and one and four one I can convert one into the other one very simply because look look what happened here one four three I've got one two three so it seems if I can change let's put something else out here let me put it down here one two three four five and one two three four five and we'll call this sigma and sigma took one to four it took two it took four to three it took three back to one it took two to five and it took five it took five to two there we go and I want to change it into that now if you look at this one and you look at this one it seems as if we can just take two to four I'm going to say one two three four five on this side if I would just be able to take two to four and four to two and if that happens two and four I just need to swap that two and four round then this one becomes that one no problem so one will stay at one three will stay three and five will stay at five and let me just do it this side and you'll see now I'm going to do it this side as well so one stays with one two goes to four three stays with three four goes to four and five goes to five that seems to be the only the thing that I need to change to do this and the reason why is I don't have to change where my cycle ends because that's a three and that's a three and that's a two and that's a two that's all I need to do and I'm going to call this permutation tau and it's there again I mean I've just written it out in reverse and what we can see now is if we do this tau which is just going to be one of the elements in s5 if I turn it into tau's inverse what does the inverse do well it just turns the arrows around and now they go that way so if I go from here to here that would be tau inverse this is still sigma and this is still tau so if I have tau composed with sigma composed with tau inverse remember how we do that we do the last one first then this one then this one it will just be as if I go along this way so if I do this in this direction that would be tau inverse there's sigma and there's tau and let's see if I can land up there if I do that so let's see what happens to one one goes to in the inverse doing tau inverse first one goes to one goes to four goes to two so one goes to two that's nice I'm getting there what happens to two two goes to four to three to three it goes to three hey it's getting there what happens to three three to three to one to one goes back to one that works what happens to four four goes to two goes to five goes to five four goes to five and then build five we'll have to go to five where are we five goes to five inverse to two to four so five goes to four so four goes to five and five goes to four and there we have it I have this one and look what I did here I did conjugation I did conjugation and now think of the conjugacy class what is the conjugacy class here of sigma if I were to do sigma remember how we do that that is I'll say it's tau composed with sigma composed with tau inverse for all tau being an element of in this instance whatever my group was that was conjugacy and now you can see something which is actually the important thing here is that all the the cycle types if I look at the conjugacy class remember and this is going to partition my group my my whole group is that all of the elements in the conjugacy class will have the same cycle type because I showed you what conjugation actually is conjugation is actually changing one cycle type into another cycle type and those two are exactly the same cycle types that's all that conjugation is doing for me so it's another way to view what actually what conjugation actually does because from what we've seen here all the conjugacy classes all the elements in the conjugacy class will actually have the same cycle types it'll be a three a three two and a three two and a three two and another three two all two three two three two three two three whatever the conjugacy class might be so very nice sort of insight into into the cycle I suppose there's something new that we learned the cycle type and that all the elements in the conjugacy class will be of that exact same type