 Welcome back to our lecture series, Math 42-20, Abstract Algebra 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. This is our first video in lecture 15, which will be our first section from chapter 5 of Tom Judson's Abstract Algebra textbook entitled Permutation Groups. And so we're going to be talking a lot about permutations in this chapter. So one permutation group that we've already come across is what we call the symmetric group. And this is essentially the mother of all permutation groups. If we have some set x, this could be finite, it could be infinite, whatever, we define the set as sub x to be set of all permutations on the set x. So these are going to be maps from x back into itself that are bijections, both one to one and onto. Now we've seen previously that the composition of permutations, or what we call permutation multiplication, is itself a permutation. And so the set of permutations is going to be closed under multiplication, the identity map is a permutation, the inverse of a permutation is a permutation. We see that the set of permutations forms a group, which we call the symmetric group, again denoted as s sub x right here. Do recall that in the special case where x is the set of numbers one, two, all the way to n, we abbreviate this as s sub n. And frankly speaking, the set x doesn't really matter too much, it's really the cardinality of the set x that's going to matter the most when it comes to these things because the set x you can just think of as labels. So for the most part, we're going to be talking about s n exclusively, but you know, when when it's appropriate, we can talk about a generic set x right here. Now chapter five is all about permutation groups. What's a permutation group? Well, if you have a subgroup of s x or s n, whichever you want to call it, we call this a permutation group. Now the usual tableau notation we've been using to represent a finite permutation is extremely cumbersome. You have to write two rows, where the first is just the domain of the thing and then how things mix up. It's a very inefficient way of representing a permutation. So in this video, we're going to introduce a new simpler notation for permutations, which we call cycle notation. And to begin, we have to talk about what do we mean by cycles with respect to permutations. And so you can see definition 511 right here. A permutation sigma inside of our symmetric group s sub x, this is called a cycle of length k. If there exists some positive and integer k and elements a one a two up to a k inside of the set x such that sigma looks like the following a one is going to map to a two a two is going to map to a three a three will map to a four a four will map to a five all the way down to a k minus one will map to a k and then a k will map back to a one. And so if you think about it in terms of a picture, this is where the idea of a cycle comes into play. So a one it maps down to a two a two maps over to a three a three is going to map to something until eventually you form this cycle where everyone just kind of maps to each other in the end. And so this cycle we're going to denote in the with the following notation sigma is going to equal you're going to put a left parenthesis there and you'll close it at the end with the right parenthesis. And then you're going to list the elements in order the first element that maps to the second element which maps to the third element maps to the fourth element which maps to the last element and then it's understood by the cycle notation that the last element will map back to the first one in this cycle. So we say that two cycles which cycles are permutations we say the two cycles are disjoint if the two cycles have no elements in common. So let's look at some examples of cycles right here. So let's look at some let's look at some permutations from s7 and s6 the first one sigma this is going to come from s7 right here. So we see that one goes to six two goes to three three goes to five four goes to one five goes to four six goes to two and seven goes to seven. So what we're going to do is notice that this is a six cycle. We often we often refer to a cycle as a k cycle where k is the length right there. So instead of saying cycle of length k. So this is going to be an example of a six cycle and we can see this in the following way one goes to six and we can kind of follow the path here one goes to six six goes to two so we say one goes to six six goes to two two goes to three then three goes to five five goes to four and then four goes back to one in which case because of that we're going to close this thing off. And so this map this permutation sigma is actually equal to the cycle one six two three four five. Notice that the element seven was actually fixed in this permutation. If we look at tau for example this is a permutation and s6 I claim this is also a cycle it's actually a three cycle in this situation. You'll notice that elements one five and six are actually fixed and so the elements two three and four what's going to get cycled around. You see from here that two goes to four you see that right here four goes to three and then three goes back to two. And so this is an example of a three cycle two goes to four four goes to four four goes to three and three goes back to two. And so this is how we're going to represent these these cycles right here. Now I should mention that as displayed in this example one can start with basically any element of x and then follow its images. So like if you look at an element x to construct this cycle then you're going to go to the next one you get sigma of x then you go to sigma squared of x and then you go to sigma cubed of x and use progress along until you get to the end you know some sigma k of x. Then the next element you're going to see that sigma k plus one of x is just x again. And so this starting and ending with x is indicative that you have some cycle going on right here. And that's exactly how we're able to do this right here. So from this we can construct a cycle starting at x and we go until we basically get x again. Of course the cycle starting at x is none other than just the set. We can actually describe this very much as a set where we look for all of the images of x under various permutations pi where pi just belongs to the cyclic subgroup generated by sigma. And so this right here is going to equal the cycle containing x and everything. So yeah this is how we can actually construct it very recursively in that regard. Now of course if we had started at some other element in this cycle we would construct exactly the same cycle. It might look a little bit different. Like so for example if you look at this sigma again we might have actually started at 2. So we've been 2 goes to 3. Whoops I forgot my parenthesis there. 2 goes to 3, 3 goes to 5, 5 goes to 4, 4 goes to 1, 1 goes to 6 and 6 goes to 2. So I want you to be aware that although it looks a little bit different these two cycles are one and the same thing. You can actually kind of move things around so we often have this little convention that we like to write those small numbers first that's in the cycle. And so if you kind of cycle this through you'll show that these two things are the same thing. That's okay. So it doesn't actually matter which element you start with to form the cycle because being in the same cycle is actually an equivalence relationship on the set x. Now I should also mention that not every permutation is a cycle. These two examples were cycles but not every permutation is a cycle. On the other hand every permutation can be written as a product of disjoint cycles up to reordering. And that's actually the statement here for theorem 513. Every permutation in Sx can be written as a product of disjoint cycles. Now how do we see this? The basic idea behind the proof is the following and we're not going to give a detailed proof here but we'll see an example of the algorithm in just a second. We can construct the cycle decomposition for any permutation by starting with any element of x. So you start with your favorite element of x. We'll call it x, right? And then you then you find the next element sigma of x. You find this next one sigma squared of x. And again you do this sigma k of x. And then you get the next element sigma k plus one of x. And then if this turns out to be x you're like oh I just finished a cycle. Then you close up the cycle and then you have to look for what's left over if we take x but you take away all of these elements in that first cycle. So look for something that's still inside there. Let's say y. So we take a y and then we start a new cycle with y. So you have y sigma of y sigma squared of y. And you're going to keep on doing this until you find y again and it closes up with some maybe like sigma l of y. And then you look okay have I gotten everything inside the set x yet? If there's nothing, if there's still something left over you start a new cycle and you'll just kind of do this process recursively until you use up all of the elements of the set x. And this will give you a product of disjoint cycles. Let me show you an example of this with the map with the permutation row right here. So this is a permutation and S6 and let's see what happens. We're just going to start with the first element one. So one is going to go to two, two then goes to four, four goes to three and then three goes back to one. Since I got my starting value again, that's a cycle. So this then becomes the cycle one, two, four, three. Now let's look inside of the set x. What hasn't been used yet? We didn't use five and six at all. So let's then start a new cycle starting at five. In which case we're going to get five goes to six and then six goes back to five. And there we have our permutation. So the cycle decomposition of row here is going to be one, two, four, three and five, six. And again, any cyclic permutation of these would also be an acceptable decomposition. So you could be like four, three, two, one, six, five. These are the same permutations because it's the same cycles. You can always just kind of rotate down or up as many times as you want these cycles here. So again, we're going to usually have the convention that we're going to start the cycle with the lowest element in that cycle, just so we have sort of a unique representation. And also we'll kind of organize the cycles by the least element. So one comes before five to give us sort of a unique factorization when it comes to permutations. Let's look at some other examples here. So we'll leave before I leave row here, we might, since this is a permutation with two different cycles, we might refer to this as a four, two cycle, meaning that we have a four cycle and a two cycle together that are disjoint from one another. Let's take a look at sigma down here, which is going to be a permutation as six as well. So if we start off, start off drawing this one, I will start with one here. You get one, one goes to six, six goes to two, two goes to four, four goes to one. So we close that thing off. The next thing to do is you're going to get, well, let's try three. Three goes to three. So we just close that one off. And then the last one here is you're going to get five goes to five. And so you get something like this. Now a convention we do with cycle notation is that if you ever have a one cycle, we just omit it. We just don't even include it. We don't want it there whatsoever. So we actually would abbreviate sigma as one, six, two, four. And we call this a four cycle. Similarly, if we did the same thing for tau right here, tau, we could take one goes to three, three goes to one. So there's a two cycle, two goes to two. So since that's fixed, we're just going to ignore it. Four goes to five, five goes to six, six goes to four. And so we see that tau here is a two, three cycle. Again, we ignore points that are fixed. Now, because of this notation here, we kind of ignore one cycles. If you looked at the identity, the identity would always look like one, two, three, four, you know, all the way up to and, you know, it's just it's just in many one cycles. And so because we don't like to write one cycles, we actually just abbreviate the identity with just a one. So if you ever see a permutation indicated as one, that means that the identity permutation, and given that we often think of permutations operation as multiplicative, then it makes sense to refer to this as our multiplicative identity. And so this is how we can express every permutation uniquely in this cyclic decomposition or the so-called cycle notation.