 So, let's continue our discussion of the supremely important permutation groups. So, one of the problems that we run into with Goshi notation is it's not very concise. And while it's generally true that paper is cheap and conciseness by itself is not a virtue, it is sometimes nicer to be able to express something on one page so we can see it on the page than it is to have the same thing expressed over several pages where we have to go flipping back and forth between pages or screens. So, what if I follow the path of an element as I repeatedly apply a permutation? The path can be described using what's called cycle notation. So for example, if I have the permutation written in Koshi form looking like this, we have the following cycles. Well, I can take the element A and again think of a permutation as a replacement table. I'm going to take A and I'm going to replace it with da-da-da, A. Well, okay, nothing exciting happens there. So, this is a what we might call a one cycle and I'm taking A, I'm replacing it with A, I'm not really doing anything. It's a one cycle, it's a cycle of length one and I'm going to write this by including the entire cycle inside a set of parentheses. Well, what about the other elements? So I can take the element B and I'll start at this, element B and see what happens. So the table says B is going to be replaced with E, E is going to be replaced with D, D is going to be replaced with C and C is going to be replaced with B and that takes me back to my starting point which is why we call it cycle notation. I have cycled all the way back to where I began and this is a one, two, three, four term cycle and so it's called a four cycle and I'm going to write it by recording where I've been. So I start at B, go to E, go to D, go to C and my next step is going to take me right back to the starting point so I don't need to include that, that's implied in the notation. And so this permutation is going to be written in cycle decomposition formed by listing what those cycles are, it's A followed by B, E, D, C. Well, traditionally I don't write the one cycles, they don't actually do anything so conventionally we just omit the one cycles as implied. Now the important thing about a cycle decomposition is that anything we leave out is assumed to be left unchanged and so if I have a permutation like this I want to make sure that I identify where everything goes and then I can drop out the one cycles. So let's take a look at what this is so I'll find a cycle decomposition so if I start at A I go to C, if I'm at C I go to E, if I'm at E I go to A and so here is a three cycle and I can write that as A followed by C followed by E. Well let's see that tells me where A, C and E go, I have a cycle that starts at B and doesn't go any place so B goes to B, it's a one cycle just written as B and let's see that takes care of A, that takes care of B, that takes care of C. The cycle we haven't included, the term we haven't included is D so I'm going to start at D, go to F and from F I go back to D and so here I have a two cycle and I'm going to write that as D, F and our cycle decomposition A, C, E, B, D, F and I don't really need to include those one cycles so I'll omit them. Again it's not really wrong if we include them, this is a cycle decomposition but we can omit the one cycle and the important thing about that is if we don't see an element listed then we can assume that element is unchanged and this is particularly important if we're trying to go back to Cauchy notation so for example if I have the cycle decomposition A, D, F, B, E then I can rewrite this, this is a permutation and I can write this in Cauchy notation and again these cycles tell me where everything goes and Cauchy notation tells me where the individual elements goes. So let's construct the Cauchy notation for our permutation so this first cycle says A goes to D, D goes to F, F goes back to A so that tells me how I can construct part of the table. In Cauchy notation A gets replaced with D, D gets replaced with F and F gets replaced with A and so there's my cycle A to D, D to F, F to A and there's my first cycle. The second cycle tells me that B goes to E and E goes to B so I'll record that B goes to E, E goes to B and there's my second cycle there and there I have my permutation. Well it looks a little bit strange because as we know the alphabet song does not go A, B, D, E, F, there's actually a missing letter in there so we should for Cauchy notation put in that missing letter C, nothing happens to it so C goes to C. Now one important thing to remember about Cauchy notation is that the permutation is not this actual arrangement D, E, C, F, B, A, this is not the actual permutation. The actual permutation is replaced A with D, B with E, leaves the alone, D, E, replaces with F and so the replacement is the permutation not the arrangement and so let's say I have two permutations in Cauchy form, A, B, C, D, E, replace with D, E, B, A, C and here's a different permutation, A, D, B, E, C is replaced with D, A, E, C, B. Now if I apply the first permutation to the sequence B, A, C, E, D, it doesn't really matter what this is, this permutation says A replaced with D, B with E and so on. So B, I'm going to replace it with E, A, I'm going to replace it with D, C, I'm going to replace with B, E, I'm going to replace with C and D, I'm going to replace with A. So if I apply the permutation to this sequence, here's what I end up with. On the other hand, if I apply the second permutation to the same sequence, I get B is replaced with E, A is replaced with D, C is replaced with B, E is replaced with C and D is replaced with A and so if I apply the second permutation to the same sequence, I get E, D, B, C, A and they're the same things. So we go back to our duck principle. If it swims like a duck, quacks like a duck, waddles like a duck, it's probably a duck. These two permutations are the same permutation and one of the things that suggest is that when I write a permutation in Cauchy form, that permutation is not uniquely described. I can have two permutations that appear different in Cauchy form but they're really the same permutation and again it's important to remember that the permutation is the actual replacement and it's not the sequence of letters down at the bottom. The permutation is not D, E, B, A, C, it's replaced A with D, B with E and so on. Cauchy form does not uniquely describe a permutation and so we think hey that's great because Cauchy form is also less compact so maybe this cycle notation is better. It is more compact and maybe there is a uniqueness that goes with the cycle form. Well let's talk about that. So here's two permutations and I'll give them in cycle form A, D, B, E, C, D, A, E, C, B and let's see these permutations. So again I'll apply both of these permutations to the sequence B, A, C, E, D and let's see. So B, so in cycle form, so that first permutation B is going to be replaced with E, A is going to be replaced with D, C at the end of this cycle gets replaced with B, E gets replaced with C and D gets replaced with A. So if I apply this permutation to the sequence B, A, C, E, D, I get E, D, B, C, A and I'll apply the same the other permutation to the same sequence and so this says B gets replaced with E, A gets replaced with D, C gets replaced with B, E gets replaced with C and D gets replaced with A. So the sequence B, A, C, E, D under the first permutation I get E, D, B, C, A. Under the second permutation I get E, D, B, C, A and let's see it quacks, it swims, it waddles, these two permutations again even though they look different in cycle form they are actually the same permutation. So again cycle form is also not unique which means that the one virtue that cycle form has over Koshy form is it's a little bit more compact but it's useful to be able to go back and forth between the two forms because neither has an inherent advantage over the other.