 In the next couple of videos, I want to talk about the centralizer and the normalizer of a group. And in order to do that, I just want to start here in Mathematica. Don't worry about the code, again, that's not what this is about. I want to just return and show you something specific about one example, and that is the dihedral group in six elements. As I showed you before, you can ask for the group elements of the dihedral group in six elements, and it gives you these 12 permutations. The 12 permutations specific to this, if you take six elements, you look at all their permutations, a certain number of them, 12 of them indeed, form this dihedral group. And we see the cycles, this is how it is expressed in Mathematica, how the permutations are cycled, and we see the first one's empty, that just means the identity element. And I've listed them here for you 1 to 12. So there's the first cycles, the second one, the third one. So the second one is 2 goes to 6, 6 goes to 2, and 3 goes to 5, and 5 goes to 3. In the parlance that we've used, the notation that we've used up till now, we would have called that tau 2635. And you see the third one here is 1 to 2, 2 to 1, 3 to 6, 6 to 3, 4 to 5, and 5 to 4. That will be tau 1, 2, 3, 6, 4, 5. The fourth one here would just be sigma. 1 goes to 2, 2 goes to 3, 3 goes to 4, 4 goes to 4, 5 goes to 5, 6 goes to 6, 6 goes all the way back to 1. That was the sigma. And we see all 12 of them here. We sigma twice there, sigma 3 times, sigma 4 times, sigma 5 times, and we see all these other ones. So that's good and well. We know what this is about now. We've seen the dihedral group. We know the 12 permutations that are included in this dihedral group in six elements. And let's just consider one of them. I'm just picking an arbitrary one. Let's pick tau 1346. That will be the fifth one here. 1, 2, 3, 4, 5 there. 1346. And what I want to do, I'm going to run through all of these while I'm going to skip the identity element because it doesn't do anything interesting. But I'm going to go through all the elements in G and I'm going to see if they commute with element 5, if they commute with this tau 1346. So I'm going to check if G composed with tau 1346 equals tau 1346 composed with G. Do they commute? Very easy in the Wolfram language. I just write a little table. I cycle through 2 to the end and I know it's the end because I want the length of all of these and because they're 12, it's going to go from 2 to 12. I'm going to cycle from 2 all the way to 12. And what I'm going to do is I'm going to say, so take 2 and 5 and that is that equal to 5 and 2. And then it's 3 and 5 equal to 5 and 3. And then it cycles through all of these and I get 2 false values back. And as you can see there, I get false, false, false 2. And that is the 2, 3, 4, 5. So indeed 5, that 5 tau 1346 is going to commute with itself. Tau 1346 composed with tau 1346 equals tau 1346 composed with tau 1346. That's 2. So 5, 6 doesn't, 7 doesn't, 8 does. 8 was sigma 3 and that would commute with 5. So 8, 9, 10, 11 doesn't, 12 does. So this last tau, if I take tau 162534 and I compose that with tau 1346 and I commute that. Those 2 are equal to each other. So there's certain elements here that commute with this. And once again, very nicely in Mathematica, I can just ask for the centralizer. Because these elements, I take one out because the identity element would be there. It does commute and it will commute with 5. So it's actually 1, 5, 8 and 12 of these. So 1, that one, that one and that one, they all commute with 5. And I just chose 5 at random. What you could also do is just choose a random subset of these. And then go through and see which ones commute with each ones of those. And that will be special ones. And we call those the centralizer. And in the next video, I'll show you exactly what a centralizer is. But you can clearly see there is something special. If I take one of the elements or 2 or 3 or 4, some subset of my group. And I can check which of all the elements in the group will commute with those. And that is called a centralizer. And as I say, I can be nicely asked for the group centralizer for the dihedral group. And the ones I want to check out in this instance is not a subset, just a single one. Well, a single one is a subset. And it shows me there the cycle, the cycle, the cycle, the cycle. Which would be the identity element, which would be itself. Which would be 8, which is sigma 3. And which would be 12, which is tau 162534. So I can create this V nice. This is a very nice subset of my initial group G. Which would commute with this new element that I chose from there. And from that, I want, in the next video, we're going to build our understanding of what the centralizer is. And in the video thereafter, just a bit, just relax things a little bit. And we're going to look at the neutralizer. So that is what is ahead.