 So let's play around a little bit more with cycle notation and see what comes of it. So one of the things that we want to be able to do is compose two permutations. And so if I have two permutations again, remember that we indicate the operation of composition by juxtaposition sigma tau, where this is read as apply the permutation tau first, then apply the permutation sigma. So if I have two cycles or more, I can also indicate composition in the same way. I could put them right next to each other and that says perform this permutation first and then do this permutation. Now as with permutations, again, we go from right to left, as with permutation. Again, it is important to remember that the permutation is not the ordering of the letters, but it's the replacement of the letters. So if I want to think about this, maybe I want to form the Cauchy table for this permutation. So let's see. So A, what is A going to be replaced with? Well, I'm going to apply this permutation first and because this one doesn't have A in it, I can assume that nothing happens to A, but this one says A is replaced with D. Let's see. So now let's see the letters of the alphabet, A, M, N, X, now A, A, B, so B is the next letter. So B, my permutation says don't do anything to B, don't do anything to B, so B stays the same. Next letter C, don't do anything, don't do anything, so C stays the same. Next letter D, D gets replaced with E, but then E gets replaced while nothing happens to it. So here's an important thing. The first cycle here replaces D with E. I do have to apply that second cycle and nothing happens to E. E, first cycle, replaces E with D, but wait, D then gets replaced with F. And finally, F, nothing happens, F gets replaced with A. So again, as with other permutations, composition is not necessarily commutative. So if I reverse the order of these cycles, well again, I'll do the same thing. So A gets replaced with D, but wait, D then gets replaced with E. So in the composition, A is going to be replaced with E. B, again, nothing happens, followed by nothing happens. C, nothing happens, followed by nothing happens. D gets replaced with F by the first cycle, and then F, nothing happens to. E, nothing happens. E is replaced with D, and then F gets replaced with A, and then nothing happens to it. So here's my reversal of those two cycles. And I get a different permutation. And this creates a big problem with our notation. If I write a cycle decomposition like this, A, C, E, D, F, what this might mean is it might mean a single permutation that looks like this. Maybe when you see this, it means a permutation that's been decomposed into cycles. But maybe I'm juxtaposing two cycles. This might mean the composition of two cycles. So this cycle, D, F, looks like this. Maybe I apply this permutation first, and then this cycle looks like this. Then I'll apply this permutation. Do I mean this or do I mean a composition? And this is very bad because it means that a juxtaposition of two cycles could mean two very different things. So which one is it? We don't like our notation to be ambiguous. Well, before we panic, let's see how big of a problem it is. So let's say I do have an expression like this, A, C, E, D, F. For example, this is a cycle decomposition. So if I read this as a decomposition of a permutation, then that permutation is going to be, well, A goes to C, C goes to E, and E goes back to A, and then D, B, nothing happens to you. D goes to F, and then F goes back to D. So this cycle decomposition corresponds to this permutation. On the other hand, if I read this as the composition of two cycles, then it's going to be the permutation. So let's see. So what's going to happen? Well, A, the first permutation does nothing to it. The second one, A goes to C. B, first permutation does nothing, followed by do nothing. C, do nothing, followed by go to E. D, first permutation, goes to F, do nothing. E, do nothing, followed by go to A. F, replace with D, and do nothing. So if I read this as apply this cycle first, then apply this cycle, then this permutation is going to give us that. And so now I have to decide, do I, when I write this, do I mean this permutation, or do I mean this permutation? So let's see, thinking, thinking, thinking, thinking. Which one do we want it to be? Wait a minute, these are the same permutations. They look exactly the same and they happen to be the same permutation. So maybe this is not that big of a problem. It suggests, in any case, that we can read a juxtaposition of two or more cycles, either as the decomposition of a permutation or as the composition of cycles, because either way we read it, it looks to be the same. Except there's another problem. Composition is not commutative. If I compose two permutations, it's not going to, in general, give me the same permutation. So it's possible that the cycle decomposition of a permutation may actually depend on how we find it, and that if I reverse the order that I list the cycles, I may end up with a completely different permutation. Fortunately, we do have the following result. If I have two disjoint permutations, not just cycles, but in general, any sort of permutation, if I have two disjoint permutations, then the composition in one direction is the same as the composition in the other direction. In other words, if my permutations are disjoint, they commute. And this is something you should actually prove on your own, as this is an important result for the algebra of permutations.