 So we're going to move on to a new topic but before we do that I want to expand on the little example that we've just seen where we used what is actually a dihedral group. So I want to get to a new topic but first let's stick to these dihedral groups. Remember the example that I used in the previous video just the four elements and we transformed that. So when we get to this dihedral group we usually donate a D and we have a little subscript N and that N is an element of the natural numbers. So what's going on here? This is where we have this interesting idea where we take something that we can actually look at and we transform it by flipping it or we rotate it and what we want that object is to look exactly the same as we did before. So we had our little square and we had numbers to 1, 2, 3, 4 but what we want to do to that is to rotate it or flip it along any of the diagonals so that if I didn't put numbers to that it was just that little square. If you looked at it and you looked away and I changed it and you look back again you notice no difference. That square looks exactly the same. Now we've marked them 1, 2, 3, 4 so we can see that had moved but when you just look at that square without the numbers you don't see anything. Nothing has changed and that's what we talked about the dihedral group. And they're very interesting when N equals 1 and N equals 2 but when it gets to 3, 4 and more then it gets more interesting. And what we are dealing with here is just as well we're dealing with permutations. So we're going to do the permutations on a number of elements and every permutation we're going to give a symbol to remember we say sigma and we say tau. So every permutation we give a symbol to and some of those symbols are going to make it into a group so when we had those symmetric groups all the permutations make it into a group. When we have a cyclic group or cyclic group it's only the cyclic permutations that make it into a group. With the dihedral groups it's slightly different and I want to show you how we do this. Now let's just start with D1. So if we want to imagine something let's not stick to a square because a square has 4 sides let's just stick to something that's a bit easier to see. And note in that it's just a circle and we have one there. There's just one element and if we you know if we permute one it's just one goes to one. One goes to one there's just the identity element. And the way that I can do that through symmetry is if you look at this what I could do is I could rotate it through 2 pi radians full 360 degrees and it will look exactly the same or if I were to technically if I were just to flip it like this it would stay the same. But one is this map to one that's all I have and that's not very interesting. So let's look at D2 so again with this and I have one there and I have two there. So what I could do if you looked away is consider this line of symmetry here and I could just flip it this way around. And if the numbers one and two went there you wouldn't notice that there was any difference. And what would happen to the one it would stay by one and two would stay by two. So one would go to one and two would go to two and that would just be an identity element. But I could do something else I could rotate it through pi radians 180 degrees. If I were to do that then I would have one go let's do this one to one and two to two. So one would go to two and two would go to one and we'll just call that tau 12. One goes to two and two goes to one and a cycle we would just write it like that. This cycle here would just be one. So here we would have two elements we would still have the identity element and that is where we just do this flipping here this transformation. But I could also just rotate it through I could rotate it through pi over or pi radians 180 degrees and I get this I'm just going to let's just keep a tau because there's just this one. So there are these two elements and it is a group you can go through there is closure there is associativity there is an identity element and each one of these are their own inverses. If I tau and compose with another tau it's just going to give me the identity element. Identity element composed with identity element is going to give me the identity element. When we get to D3 things become a bit more interesting. So let's just have us one two three one two three. So what we can do here I mean we are going to have the one two three one two three and that is just the identity element. We can rotate it through so we take 360 we take 360 degrees and we divide it by three. That's one suddenly I can't think 120 degrees we rotate by 120 degrees so one goes to two two goes to three and three goes to one. So we'll have this one goes to two two goes to three and three goes back to one and what we'll call that is we'll call that sigma. If I go the other way around or I do this twice I'm going to give one two three three two and two to one. So one is going to go to three three is going to two and two goes back to one and we call that one sigma squared remember. Now I have this line of symmetry here and if I flip this way around one stays with one and two goes to three and three goes to two so that's another cycle. If I did this line of symmetry here three stays where it was and one or two would flip. So one goes to two and two goes to one that's that if I did this line of symmetry here we'd note that one goes to three and three goes to one so that was the cycle one three and that was nothing other than what we had let me write this out uniquely here. So the the tahedral group of three elements remember that had that would be the identity element. We had tau one goes to two two goes to one we are tau one goes to three and three goes to one and we had tau two three and then we had sigma and we had sigma squared. So certainly something we've seen before and we can make this into a group by group composition there and we've seen before that if we were to do this so this is actually just a triangle what I can do with this equilateral triangle and if I have one two and three and I would rotate them or I would use these lines of symmetry here if I were to have these lines of symmetry and I would reflect them there you know this is exactly and you looked away you wouldn't know nothing has changed about that triangle and we've seen this one this example very neatly and now we can just call it what it is it's a tahedral group in three elements and then we got the example where we looked at the tahedral group in four elements the tahedral group in four elements and this is what we had here and we saw all the rotations and reflections that we could do to do that. The next one I want to jump to is not D5 but D6 and the way that I'm going to show you D6 is just using a computer program and so I'm going to record my screen and talk you through what happens to D6 because when it comes to D6 it's probably for me the first interesting one because D4 is a bit small because what we are after in India is the center the center of a group and if we look at D6 for instance we could work out what the center of a group is and in the next video we'll start looking at that and see if we can find the center of the tahedral group in six elements. Here I am in my web browser and I've typed in the URL www.wolframcloud that's w-o-l-f-r-a-m-cloud.com and if you go to Programming Lab if you hit that you can open a free account and you can code in the Wolfram language right inside of your browser so you can go ahead and do that you can also of course use a desktop copy of Mathematica which you can either purchase or if you have an institution, a university or your work where you might have a site license to use to get a copy for free but otherwise go to Programming Lab then open a free account and you can code right inside of a browser, it's wonderful to do so let's have a look at these tahedral groups so this is what is called a notebook you would see almost exactly the same thing if you were to do this in a browser and I can just type normal words and sentences there etc and I've made a little title there so what I'm after is the group elements so I'm going to say group elements and I'm going to pass an argument to it and arguments all go inside of what is called the square brackets and I'm going to say the tahedral group I see it coming up there, I can just hit Tab to autocomplete and I want 1 and 4 elements so open and close with square brackets the 4 that is an argument passed to the dihedral function and then close the square bracket again for the group elements simple as that and shift and enter and we see all the cycles and we note that there are 1, 2, 3, 4, 5, 6, 7, 8 cycles and you'll see there is a pattern to it the number of elements these are now all going to be elements inside of this tahedral group and it will have 2n here n is 4 so it's going to have 8 it's going to have 8 elements and we were going to name all of these now the way that Mathematica works or the Wolfram language cycles with nothing in it that just refers to the identity elements so 1 is going to go to 1, 2, 2, 3, 2, 3, 4 to 4 and we see another cycle there and that is where 2 goes to 4 1 stays as 1 and 3 stays with 3 and we see another cycle 1 goes to 2 2 goes to 1 and 3 goes to 4 and 4 goes to 3 we see a cycle where 1 goes to 2 2 goes to 3 3 goes to 4 and we see a cycle and what we would do is just to name these and we'll have an identity element and we'll have some sigmas and we'll have some towels, etc and that is a very easy way if it starts getting big to just write a single line of code like this and to get these answers now the next thing I want to do is just to show you the Cayley table of this so let's have table form so I'm going to say table form and I want the dihedral group the dihedral group in 4 elements and I'm just going to use what is called post-fix notation don't worry about that this is not to teach you how to code so I want the group multiplication table and I want some table headings and we're going to make the table headings just automatic so what that is going to do for us is the following and there we go we see the 8 elements now it's going to number them 1 through to 8 instead of using sigma and because you can actually use you can use any symbols you want I mean there are some symbols that are used commonly in textbooks and by your lecturers but the Wolfram language and Mathematica here is going to name them 1 through 8 or these 3 elements that we have in our group and we see the composition so we see the composition in this order that we see them here 1 composed with 1 is going to give you 1 3 composed with 3 is going to give you back the 1 and you've got to go look what the third cycle was and whatever you named it so that is such a beautiful thing to do so let's look at this group element let's look at the group elements of the dihedral group the dihedral group with 6 so remember there's got to be 12 elements in this and we see all the cycles there and I'm just going to copy and paste so copy and paste there and let's just change this to 6 so let's just look at the group the group composition table there and we see the 12 elements in the group composition and it is from this that we're going to try and look at finding if at all possible the center we're going to try and find the center of this dihedral group with 6 elements so there you go that is the dihedral group it's actually very nice, fantastic this sort of thing to start playing with perhaps have a look at some more from language code and see if you can play around and create some of these on your own