 Suppose p is a set of n elements. For convenience, we'll call these elements 1, 2, 3, and so on up to n. We define a permutation on set p is a 1 to 1 function sigma that acts on the elements of p. For example, suppose our set is 1, 2, let's find all possible permutations on p. So our permutation will be determined by its effects on the elements 1 and 2. So one possible permutation is sigma, and we'll decide what sigma does to 1 and sigma does to 2. And so maybe sigma sends 1 to 2, and because it has to be a 1 to 1 function, this means that sigma 2 has to be equal to 1. Now if I want another permutation, let's call it rho. Again, rho will do something to 1 and do something to 2, and if I want it to be different, I'll let rho send 1 to 1, and that means rho sends 2 to 2. Could there be more? If we try to find a third permutation, tau, then either tau of 1 is equal to 1, which means that tau of 2 must be 2, and that's the same as rho. Meanwhile, if tau of 1 is equal to 2, then tau of 2 has to be equal to 1, which is the same as sigma. And remember, things that do the same thing are the same thing, so any other permutation, tau, is either rho or sigma. Now, given a permutation sigma, we can express it compactly using what's known as Cauchy-2-line notation. Rather surprisingly, this is called Cauchy-2-line notation because it first appeared in an article by Augustin-Louis Cauchy in 1815. Cauchy-2-line notation consists of the following. The first line consists of a listing of all elements of our set P, and the second line consists of the images of the elements under our permutation sigma, and the whole thing is enclosed inside a set of parentheses. Here's how Cauchy wrote it. Notice that he doesn't list the elements in order, they're just any convenient listing. What's important to realize here is that even though this looks like a matrix, there are only so many symbols, and it's important to understand from context that this is actually a permutation. So let's take our two permutations on the set of two elements, and we'll express them in Cauchy-2-line notation. So our first permutation, sigma 1, is 2, and sigma 2 is 1, and so we can express this permutation by listing the elements 1 and 2 in our first row, and their images 2 and 1 in the second row. For the second permutation row, again, we'll list our elements 1 and 2, and their images in the second row, and so we write. Now, if we have two permutations, sigma and tau, we'll define the composition sigma and tau. The permutation produced when we apply tau, then apply sigma. Or, in function notation, if x is an element of the set being permuted, then sigma tau of x is sigma of tau of x. For example, if I have these two permutations, let's find sigma tau and tau sigma. So, sigma tau, we'll write that first, we'll list our elements 1, 2, and 3, and then see what happens to each element under the composition of the two permutations. So we want to find sigma of tau of 1, sigma of tau of 2, and sigma of tau of 3. Tau of 1 is equal to 3, tau of 2 is equal to 1, and tau of 3 is equal to 2. And so we find these values sigma 3, sigma 1, and sigma 2 to be. And so the second row of our two line notation is going to be the outputs 3, 2, and 1. Meanwhile, tau sigma is going to be tau of sigma of 1, tau of sigma of 2, and tau of sigma of 3. And we find, and that gives us tau sigma. And it's worth noting that these are different permutations. Sigma tau is not the same thing as tau sigma, and that's a consequence of the fact that in general the composition of functions is not commutative. And this allows us to define a group. Let P be a set of n elements, and let capital Pi be the set of permutations on the n elements. Then Pi composition forms a group where our binary operation is function composition. Note that this is a theorem, so you should prove it, and you should do your own homework. We call this the symmetric group on n elements, and write it as Sn. We'll take a closer look at the symmetric group next.