 So we're still after the center of a group, we've looked at the dihedral groups I just want to show you the dihedral group in six elements as we promise because we're going to find the center of this massive group. So if I stand the side there you can see my six-sided polygon and I've named it one two three four five six and the six points here on the side the corners and let's do a bit of swapping the surround so that when I do something to it and you look away and you look back apart from these numbers when these numbers gone this figure stays exactly like like it is now. So number one is very simply we do nothing one stays at one two two and two three stays in three no problem there. Let's rotate it by 60 degrees if I rotate it by 60 degrees you're not going to know the difference if these are not there but one is going to go to two two is going to go to three three to four four to five five to six six to one so that is going to be that permutation and one goes to two two goes to three three goes to four four goes to five five goes to six six goes back to one. So that's one rotation through 60 degrees but let's rotate it twice and we'll call that sigma squared so one is going to go to three two is going to go to four three is going to go to five four is going to go to six and six is going to go to two so let's see what that happens there one goes to three one goes to three so one goes to one two three goes to one two three goes to five and five goes back to one so we have to close that and then two two is going to go to four which is going to go to six which is going to go back to 2. So we see that cycle. Now we can do it three times. So that's 180 degrees pi radians that we're going to go 1 goes to 1, 2, 3. 1 goes to 4. 2 goes to 5. 3 goes to 6. And we rotate it through 180 degrees and we see there. Now we can rotate 1 all the way to 5 and we can rotate 1 all the way to 6. And we see it there. And if we do all the way to 6, 1 goes to 6, 6 to 5, 5 to 4, 4 to 3, 3 to 2, 2 back to 1. So that's one thing that we can do. But what about just flipping these? So let's flip them. Let's flip it along this axis here. So 1 is going to stay with 1. 1 is going to stay with 1 and 4 is going to go to 4. But 2 and 6 will swap and 3 and 5 will swap. So we get 2 and 6 will swap and 3 and 5 will swap. So that's along this axis. If we fix these two points and flip it along this, 2 and 5 are going to stay. 2 and 5 are going to stay. But you can see what's going to happen to 3 and 6 and 1 and 4. And then finally we can do along this axis. And 2 is going to go to 4. And 1 is going to go to 5. 5 goes to 1 and 4 goes to 2. But there are three more lines that we can draw through this and we can rotate. If we take this axis here and we go 90 degrees up against it right there, if we flip that we see 1 and 2, 3 and 6 and 4 and 5 are going to go. 90 degrees along this one, which means here, I flip it here 1 and 4, 2 and 3 and 5 and 6 are going to go. And 90 degrees along this one, in other words down there, we get 3 and 4, 2 and 5 and 1 and 6 that are going to go. So we see, as we expected, we see these 12 elements that we can make up to n, nb6. We see these 12 specific permutations that we can do in this dihedral group. And this is a dihedral group with even number of elements in the set. Even number of elements in the set. And when we have a dihedral group with even numbers, the following is going to happen. We're going to see the center. Just remember this for now because we haven't discussed what the center is. The center is going to be in the center. We're going to have the trivial rotation there. Nothing happens. And the 180 degree or pi radians rotations. So the one where 1 goes to 4 and 4 goes to 1, 1 goes to 4, 4 goes to 1, 2 goes to 5, and 5 goes to 2, 3 goes to 6 and 6 goes to 3. These two are going to be very special. So if we have dihedral group and even elements, the identity element and the 180 degree rotations are going to form what we call the center of a group. And I'm going to go quickly to Mathematica and I'm just going to show you a screenshot of that so we can have a look at these two very specific. What happens to these two very specific? If we look at the group, so that is the dihedral group and six elements, that is this dihedral group and six elements, it's set in this binary operation of just combining any two of these with the binary operation there. And we're going to see something very special. Of course, that one, it's the trivial one. There's always something special about it, but there's something very special about this 180 degree rotation as well. Let's go have a look. So there we go. We see Cayley's table as we did before for the six elements, the six, the dihedral group and six elements, which gives us these 12 permutations and Mathematica here has just numbered them one to 12. But one would be the identity element and I want you to have a look at element number eight here, which is actually the rotation through pi radians. So one goes to four and four goes to one, two goes to five and five goes to two and three goes to six and six goes to three. Have a look at this. The identity element will always commute because composing with anything or anything with it in a set that is always going to commute. But have a look at this rotation here, which Mathematica is called eight here. Just take my word for that. This is the rotation through 180 degrees. If I compose eight with one, I get eight and if I compose one with eight, I get eight, obviously. But have a look at this. If I compose eight then with the next element, which is two here, I get seven and if I take two, composed with eight, I get seven as well. And eight with three is nine and three with eight is also nine. Ten, ten, twelve, twelve, eleven, eleven, two, two, one, one, three, three, four, four, six, six, five, five. There's commutivity here. There's something very special about this element, the identity element and the rotation through 180 degrees, that element, that permutation, they commute with all others and these two elements, they form a subgroup. We're going to show that, prove that that is a subgroup of G and that is actually then those two elements is what is called the center of G.