 In this video, we're gonna talk about conjugates of permutations inside of the group SN. So imagine we have two permutations, sigma and tau that belong to SN, and suppose we take some letter A that sits somewhere between one, two, all the way up to N, right? So then, when you take the conjugates, so you take tau conjugated by sigma and sigma inverse, this is gonna be itself a permutation, now we know it belongs to SN, but it's gonna send sigma of A to sigma of tau of A. In particular, if tau is a K cycle, so A1 goes A2, A2 goes A3, all the way up to AK, which will wrap back to A1. In that situation, if you conjugate K cycle tau by sigma, this will produce the K cycle, sigma A1 goes to sigma A2, which goes to sigma A3, all the way up to sigma AK, will wrap around back to sigma A1. So if you conjugate a cycle by a permutation, basically just take all the numbers that were in the cycle and then you just look at our images with respect to the permutation sigma right here. Now, the reason why this is super relevant here is that furthermore, two permutations are conjugate if and only if they have the same cycle structure. So two permutations are conjugate if and only if they're cycle structures at the same, it's pretty cool here and we'll see an example of that just in a second. Now, the first statement is actually fairly immediate. It really is insane much, honestly. So if you take sigma tau, sigma inverse, well, okay, what does that do to the element sigma A? Well, sigma inverse and sigma will cancel out so you get sigma of tau A. So what does the function sigma tau, sigma inverse do to the element sigma A? Well, it sends it over to sigma of tau of A just like it said right above here. So again, that statement's kind of a triviality. It doesn't really say much when you think of it that way, yikes. But it's getting us in the direction we need to go. We're gonna take one step at a time. So let's take this first statement and apply it to a K cycle, right? So K sends, you know, A, sorry, excuse me, tau sends A1 to A2, which sends that to A3, which then goes in this pattern all the way down until we get to AK, which then wraps that back down to A1. That's how this cycle structure works here. So if we take sigma of A1, which the index I, we're gonna think of working at mod K in this situation here. So we know that sigma tau, sigma inverse of sigma of A1, this will map over to sigma of tau of A1. That's just what the first statement says. But what does tau of A1 do? Assuming A is inside of this cycle right here, which it is, it's the AI. Tau is just gonna send AI to AI plus one. Again, when you reduce that index mod K right here. And so then we see what happened when sigma of AI just mapped over to sigma of AI plus one. That's what this product, sigma tau, sigma inverse is doing. So sigma A1 will move to sigma A2, which then moves to sigma of A3, all the way down to sigma of AK, which will then wrap around back to sigma of A1. Okay, so the good takeaway here is that if you conjugate a K cycle, you get back a K cycle. So that didn't change, right? And so that's gonna be the, that's one of the important observations here. So when you conjugate a K cycle, you get back a K cycle. Well, what if you take a general permutation? Every permutation would be factored into a product of disjoint cycles. And so really what I wanna show you here is next is that conjugation is very compatible with this product. What happens if you conjugate a product of two permutations, say row and tau, which we make no assumptions about row and tau, they're not necessarily cycles. They could be, they could be, we don't know. But what you can do is you can insert a strategic identity in here, right? So I guess I'll just call it one, right? So if you take sigma row tau, sigma inverse, you can insert an identity in there. So sigma row the identity, right? One tau, sigma inverse. But what identity we wanna use, we actually wanna use sigma inverse, sigma, tau, sigma inverse here. So that's gonna be nice here because you have this identity we've inserted in here. But if I do the parentheses in a slightly different way, you'll notice you have the conjugate of row and the conjugate of tau just like you see right here. And so if you have a product of permutations, you can actually consider each of their conjugates separately. And so take your permutation in question right here. It's a product of disjoint cycles, okay? So you have like your permutation, we'll call it tau here. So you have like cycle one, which we'll call tau one. You got cycle two, tau two, all the way down to say like tau r, however many cycles you have there. And so then if you want to conjugate that, you're gonna get sigma tau, sigma inverse. This will become sigma tau one, sigma inverse. That's the first one. Then you're gonna get sigma tau two, sigma inverse. That's your second one going all the way down. In which case then you're gonna get sigma tau r, sigma inverse. And so when you take the conjugate of a permutation, you can take the conjugates of each of the individual cycles for which each cycle like we saw before, it's conjugate will still be a cycle of the same length. So then there exists, we see that there's, so that tells us that if you take the conjugate, the cycle structure will be the same. So that's the first direction. We wanna show that the conjugates definitely if they have same cycle structure. So the conjugates will have the same cycle structure. Well, if you have two permutations with the same cycle structure, so you have like another permutation row, exact same cycle structure. You got like row one, row two, all the way down to row r. And you know, these are K one cycles. These are K two cycles. These are up, these are K r cycles, something like that. And then you're gonna have a bunch of numbers in this one and a bunch of numbers in this one. And so we can make a bijection going between those. And so you have the numbers in this one, the numbers in this one, you make a bijection. Just going through the list, you're gonna be able to correspond all of these numbers together. So let me show you an example to make this a little bit more explicable we have here. So let's say row is the permutation one, two, three, four, five. So this would be like a permutation living inside of S five. It's a two, three cycle. And then suppose for another example, tau is, oh, we could just make up something, right? We'll just make it to be two, five and one, four, three. Right? So it's another two, three cycle. So what we can do is we can then construct a bijection between the letters in row with the letters in tau. And how is that gonna turn out? It's like, okay, one, two, three, four, five. What does sigma do? So we're gonna get that one goes to two, two goes to five, three goes to one, four goes to four, five goes to three. Just record those down. One goes to two, two goes to five, three goes to one, four goes to four, and three goes to five, like so. For which we might want to write this as using the cycle notation. So one goes to two, two goes to five, five goes to three, which goes back to one, four is fixed. So that's the permutation that connects these two things together, all right? So that's what we're trying to talk about here with this bijection. So that bijection we're gonna call it sigma, okay? Well, then we see that by the construction we have here, by the construction of sigma, we see that sigma rho sigma inverse is gonna equal tau. And that follows from our observations right up here, right? That when you take this thing, it's gonna connect the things together. That's proving what we were trying to argue here. That if you have the same cycle structure, then sigma, this bijection will actually give you the conjugator you need. And then let's just check it for this example here. If I take sigma rho sigma inverse, so sigma is one, two, five, three. I forgot the one there. So rho is gonna be one, two, and three, four, five. And then sigma inverse, I was right backwards, so three, five, two, one. And so you can see what happens here. So if I look at what happens to two, two goes to one, one goes to two, and then two goes to five. So we end up with two goes to five. So then we go through it, five goes to two, two goes to one, and then one goes back to two. So that finishes that cycle off. So the next cycle here, we're gonna get one goes to three, three goes to four, and that's it, one goes to four. Four goes to five, five goes to three. And then sure enough, three goes to five, five goes to three, three goes to one. This in fact then gives us tau. Now we're trying to argue here that this sigma we constructed, this bijection will give you the correspondence based upon the principles we saw earlier. So in SN, the conjugates are exactly those permutations with the same cycle structure.