 In this short video I wanted to show you how you can compute the inverse of a permutation when it's in cycle notation. Let's start off with a cycle, it's pretty nice. If you have a cycle sigma is equal to a1, a2, all the way up to ak, then the inverse permutation will just be write the cycle backward. So for example, if you take sigma to be the cycle 1624, then its inverse will just be to write the cycle backwards. We get four, two, six, one. The idea is you just have to reverse the process. That's what inversion does here. If one goes to six, then we want six to go to one. If six goes to two, then we want two to go to six. If two goes to four, then we want four to go to two. And if four goes to one, then we want one to go to four. So we reverse the process. Now remember earlier though, we did mention that when we write a cycle, we like to write the smallest element first. So this one can just kind of move to the front right here. And so you'll often write this as one, four, two, six. This would be the inverse. But again, there's no requirement that you have to write the smallest element first in your cycle structure here. But the inverse is just writing the thing backwards and then move the last element to the front if you want the smallest element to be in the front. That's how we take care of cycles, inverses of cycles. What if you have a union of cycles here, a product of cycles? Well, in this situation, you use the shoe sock principle that we've learned previously that you put your socks on then your shoes, then you take your shoes off then your socks. So when you take the inverse of a product, you'll reverse the order. So tau inverse, which is a two cycle times a three cycle, the shoe sock principle says that this is gonna be the inverse of four, five, six times the inverse of one, three, okay? For which then you turn these things around. So the inverse of four, five, six will be six, five, four and then you're gonna get three, one right here. Now, if you want again to put the smallest element first, you can do that. You get four, six, five and you get one, three. Oh, look, two cycles are their own inverses. And then the other thing I should mention is that, oh, wait, disjoint cycles actually commute with each other. I could write this as one, three times four, six, five. So I could still follow the convention that the smallest, that you order the cycles by the smallest element in front, right? So in fact, if you forgot to use the shoe sock principle here, it turns out it's okay because you had a disjoint product of things. Now, this is only if it's a disjoint cycle decomposition. If you just had like two permutations, one, two, three and then you had say like three, four, five, notice in this situation, these are not disjoint. So if you want to find the inverse of this, you definitely need the shoe sock principle. So you're gonna get five, four, three times three, two, one and you can reorder them if you want to. But the order gets swapped around when you have to take inverses. That's an important thing to remember here. And so that ends our lecture here, lecture 15 about cycle notation of permutations. If you have any questions, feel free to post them in the comments. If you learned something, give a like, push, whatever. Subscribe if you wanna see more videos like this in the future, I'll hopefully see you next time in lecture 16, which we're gonna talk about a very important subgroup of the symmetric group known as the alternating group.