 So let's talk about a different way of representing permutations, and this comes from the following idea. Let sigma be any permutation on n elements. The forward orbit of an element x is the sequence x, sigma x, sigma squared x, and so on, where we're using function notation, so sigma squared is sigma applied to sigma applied to x, and so on. Since the number of elements is finite, the orbit eventually repeats itself. And the distinct elements in the orbit form a cycle, and if the repotent, the repeating unit of the orbit, is the elements k1, k2, and so on, up to kn, we call it an n-cycle, and can write it in cycle notation as this very complicated expression where we list the elements in the repotent. So let's try to find a cycle. And suppose we start with the element 1. So we'll apply our permutation to 1 and get 2. Then our permutation applied to 2 gives us 4. Our permutation applied to 4 is 3. Our permutation applied to 3 is 1. And at this point we're back where we started, and so the elements will repeat, and the sequence 1, 2, 4, 3 forms a cycle, which we can write in our cycle notation. Now if we start with 1, the permutation produces the cycle 1, 2, 4, 3. But what if we started with 2? Starting at 2 gives us the forward orbit, and so our cycle would be 2, 4, 3, 1. Now the cycle has the same elements in the same order, but it starts at a different point. And the question you've got to ask yourself is, does it matter? Well let's consider. Let's take three cycles, and let's see which of these represent the same permutation. And because there are four elements listed in the cycle, we'll assume they're acting on four elements. So let's take a look at that. The cycle kappa 1 equals 1, 2, 3, 4 means that if I apply the permutation to 1, I get 2. If I apply it to the output 2, I get 3. If I apply it to 3, I get 4. And if I apply it to 4, I get 1. And so this means our actual permutation in our Cauchy 2-line notation would look like this. This second cycle kappa 2, 3, 4, 1, 2 means that 3 goes to 4, 4 goes to 1, 1 goes to 2, and 2 goes back to 3. And so our permutation is the cycle kappa 3, 2, 1, 4, 3 means 2 goes to 1, 1 goes to 4, 4 goes to 3, and 3 goes to 2. And so our permutation will be. And so in our more familiar Cauchy 2-line notation, here are those three cycles. And the thing to remember is that things that do the same thing are the same thing. And so here we see that kappa 1 and kappa 2 are identical permutations. So even though they were written as different cycles, they actually represent the same permutation. And so this suggests two cycles are the same if they have the same elements in the same order with the first element of the cycle following the last element of the cycle, regardless of which element is listed first. Now a cycle is a permutation, and so we can find the product of cycles in the same way we found the product of permutations. Later on we'll see a little twist in this that makes it a little bit easier to find the products of cycles. So let's say I want to find the product of the cycles 1, 2, 1, 2, 3, and 1, 2, 3, 1, 2. And as always we should ask the question, are they equal? Now since we want to start by thinking about these permutations, the things to note is that the products include the cycle 1, 2, 3, and so this means these must be permutations on at least three elements. And so we have the cycle 1, 2, well 1 goes to 2, and 2 goes to 1, and since there's a third element, that third element doesn't go anywhere. It's not permuted by the cycle. Similarly, the cycle 1, 2, 3 tells us that 1 goes to 2, 2 goes to 3, and 3 goes to 1. So let's find the product of the permutations 1, 2, by 1, 2, 3, that's this permutation. And if we compose them in the other order we get, and notice that this product does give us different permutations. So the product of two cycles is not commutative.