 It will be useful to be able to multiply cycles. It's important to keep in mind two ideas. Everything goes somewhere, cycles are functions, and importantly, in the cycle alpha-beta, the cycle beta is performed first, followed by alpha. So we can multiply two or more cycles by starting with any element and see where it ends up. If it doesn't return to itself, we can continue the cycle, then lather, rinse, repeat until we've tracked all the elements. So let's find the product of cycles 123 by 123. Now because this product of cycles corresponds to a composition of functions, what we're really doing is we're applying a function, and then we're applying a function to the output. So we'll start with our element 1, and we'll drop it into the first function, which sends it to 2. Now we'll drop 2 into the second function, which sends it to 3. And so 1 is sent to 3, and in our cycle notation, we would follow 1 with 3. But wait, there's more. So now we're at 3, let's see where 3 goes. So we'll drop 3 into our first function, which gets sent to 1, and 1 gets dropped into the second function, which sends it to 2. And so 3 is sent to 2. And finally, we'll see where 2 goes. So starting at 2, we see that 2 gets sent to 3, and then 3 gets sent to 1. So 2 is sent to 1, which is where we started. So our cycle is 132. Now since this is all elements that could be permuted by these cycles, we're done, and so the product 123 by 123 is the cycle 132. How about something like 13123? So remember these compositions are read from right to left. So first we perform 123, and then we apply 13 to the outputs. So again, we can start with 1. We see that 1 gets sent to 2, and the second cycle doesn't affect 2. So when we drop 2 into it, 2 just stays, and so 1 is sent to 2. Again, starting at 2, we see the first cycle sends 2 to 3, and then the second cycle sends 3 to 1. And so 2 is sent to 1, which is where we started, and so our cycle so far is 12. Now since the cycles also permute 3, we also need to find where 3 goes. So we take 3, we drop it into the first cycle and get 1, we drop that into the second cycle and get 3. So 3 is sent to itself and produces the one cycle, 3. And so our product, 13 by 123, is 12 by 3. Now since one cycle doesn't move its element, it doesn't actually do anything, and so we typically omit it. What if we apply 132 to 123? So again, we read from right to left. So first we apply 123, and then 132. So starting at 1, we see 1 gets sent to 2, and 2 gets sent to 1, and that's back where we started, so we obtain the cycle 1. Now remember, everything has to go somewhere, and so our cycles permute 2, so let's see where 2 goes. 2 gets sent to 3, and 3 gets sent to 2, and so we obtain the cycle 2. 3 gets sent to 1, which gets sent to 3, so we obtain the cycle 3. And so this cycle product, 132 by 123, is actually a product of 3, 1 cycles, 1, 2, and 3. But if we omit the 1 cycles, our product does nothing. Well, we don't want to leave a blank space, but remember the do nothing permutation is the identity, and so we can write our product as 132 by 123 is e, the identity permutation.