 Welcome back to our lecture series math 4220 abstract algebra one for students at Southern Utah University. As usual I'm your professor today. Dr. Andrew Misseldine In this lecture 18, we're gonna start the very short chapter in Tom Judson's abstract algebra textbook about cosets in Lagrange's theorem chapter 6 Section 6.1. It's gonna be our cosets and that's where we're gonna spend the majority of our time in this In this chapter, we'll spend two lectures on it and then we'll talk about Lagrange's theorem in chapter And Lagrange's theorem in chapter in lecture 19, excuse me So let's first define the titular topic of this chapter here cosets. What is a coset? So a coset is our structures. There's sets a subsets of a group and so let's give us a little bit of terminology Let's say that G is a group. This could be a finite group It could be an infinite group doesn't make any bit of a difference in this definition And then we're gonna take a subgroup of G. So by subgroup we're taking a group inside of a group using that same binary operation And the restriction of the binary operation to H gives a well-defined operation on H And so therefore H is a subset of G which is closed under multiplication. It contains the identity It contains the the inverses and you're gonna see those three principles come into play in this lecture Why it's so important that H is a subgroup for the following definition So given a subgroup of a group G we define the a left coset of H With representative little G inside the group as the following set This will be denoted little G times H and this will consist of all products of the form Little G times little H where little H is allowed to vary over the elements of the subgroup H We'll see examples of this in just a second. This is an example of what we call a left coset This is the left coset of G the representative there Okay, the set of all left cosets will be denoted as G H and we do use a forward slash right there when we're describing the left cosets G G Oftentimes this is read as G mod H for reasons that will become more clear in future chapters Specifically, let's see. I think with Judson's textbook that would be chapter. Oh boy chapter nine I think sevens about cryptography eights about coding theory So I think chapter nine is when we get to quotient groups But don't quote me on that one even though it's now record for recorded for time in memoriam in this video right here So we have G mod H is the set of all left cosets Similarly, we can define What's called a right coset a right coset will denote this as G G slash or sorry save one more time H slash G This is even set of all products of the form little eights little G where G is a specific fixed element of the group We call it the representative of the coset and then little H is allowed to vary over the elements of H right there And so this is why we call it a right coset because we're writing the representative on the right side of H and As opposed to a left coset the representative G shows up on the left side of the subgroup And in general since a group is non-Abelian We will see that the left cosets in the right cosets might not match up and so distinguishing between the two is very important the set of The set of right cosets. We are going to denote H Backslash G so notice the different ordering right there So for right cosets, we write the H slash the G But for the left cosets we write G slash the H Alright, and the direction does depend on things. We have a forward slash here We have a backslash right here If you are do it using latex you want to type the right cosets here You have to be a little bit cautious because the normal backslash is actually a protected character because this is how you start commands So if you actually want to write the symbol backslash you use the command backslash Backslash, that's kind of fun, right? So you actually just spell it out backslash backslash That's how you get the backslash symbol if you need to talk about right cosets And the idea why we move things around is supposed to be analogous to the cosets themselves So you write a left coset you have some element of G times H right there And so that's what we do right here. We have a G on the left H on the right for the right cosets We do the same thing we have H Slash G where G is on the right because that's where the representatives go and H is on the left because that's where the cosets That's where the subgroups would go It's like I said, it should it should be very important that in general we do not have it It's not true that GH is equal to HG that of course is true for abelian groups And it is true for it's true for special cases and I'm not abelian group But in general, that's not a true statement. I should be saying that it's not equal I keep on saying not equal but I didn't write that on the screen So the left coset is not necessarily the same thing as the right coset. Let's see some examples of that So let's first start off with a fairly simple group. We'll take The I say simple of course in that it's not complicated not in terms of how simple groups are usually defined Check that out in a different video. Actually, what do I mean by that? But let's take the group z6 It's a cyclic group and let's take the subgroup generated by the element 3 So just as a reminder, this is that group right here H is generated by 3 You're gonna get 0 and 3. This is a cyclic group of order 2 inside of a cyclic group of order 6 What are the cosets of this group gonna look like? Well, the first idea is what you know If we go through the six elements of z6 the first one is a 0. So what happens if we take 0 plus H? Well, if you take 0 plus H, you're gonna take 0 plus 0, which is equal to 0 and you're gonna 0 plus 3 Which is equal to 3 and so notice in this situation that 0 plus H is just gonna give you the set 0 comma 3 That's of course, just the original subgroup H itself and you're gonna see that in general if you take the coset Represented by the identity of the group you would just reproduce the original The original subgroup. What if we take 1 plus H and that situation you're gonna take 1 plus 0 Which is 1 and you're gonna take 1 plus 3 which is equal to 4 So we get the elements 1 and 3 like so Because of this first observation because a subgroup always contains an identity You're always gonna have your representative Combined with your identity. So for any group under the Sun if you're looking at G H right here You have to consider G times the identity which is equal to G which shows that the coset represented by G always contains G itself So 1 plus H in our example will contain excuse me will contain H But it also contains what I write plus 3 that should be plus that should be a 4 right there 1 and 4 All right, the next one we're gonna take is 2 plus H and my similar calculation You're gonna get 2 plus 0 which is 2 and you're gonna get 2 plus 3 which is 5 Like so and so that's getting so we have three of the cosets 0 plus H 1 plus H and 2 plus H If we take 3 plus H what you're gonna see is the following you're gonna take 0 plus 3 which is 3 And you're gonna get 3 plus 3 which is 6 but as we're working on 6 that becomes a 0 Now if you're having a little bit of deja vu, you'll notice that oh 3-0 that's actually just the subgroup H which is also the coset 0 plus H So one thing I'm gonna mention here is that this cosets already on the list And so I'm actually gonna write it right here. This is 3 plus H. All right What about 4 plus H if you take 4 plus H that would be 0 plus 4 which is 4 and then you're gonna take 3 plus 4 Which is 7 which reduces to 1 mod 6 wait a second. We already have that coset listed as well So I'm actually gonna get rid of this and just record it up here 1 plus H and 4 plus H are actually the same coset And then the last one to consider is 5 plus H, but you might see where this is going here 5 plus 0 is gonna be 5 and 5 plus 3 is gonna be 8 which reduces to 2 mod 6 So 2 plus H and 5 plus H are actually the same cosets again And you'll notice here that 2 plus H will contain 2 and 5 plus H will contain 5 So it kind of sees I kind of see the connection there 1 plus H will contain 1 But 4 plus H has to contain 4. Okay And then lastly 0 plus H will contain 0 and 3 plus H will contain 3 So I mentioned earlier how the representative has to be in the coset But it turns out that when there was any overlap with the cosets, they actually had to be the exact same coset Let's look some more examples of this this time Let's look at a non-Avelian group because if we go back to the Avelian group I can also tell you that in all of these cases, you know 0 plus H will be the same thing as H plus 0 1 plus H is the same thing as H plus 1 2 plus H is the same thing as H plus 2 3 plus H is the same thing as H plus 3 And 4 plus H was the same thing as H plus 4 and likely H plus 5 plus H was the same thing as H plus 5 Because Z6 is an Avelian group the operation is commutative and there's no distinction between Left cosets and right cosets that distinction only shows up for non-Avelian groups So let's look at a non-Avelian group. Let's take another group of order 6 But this time we're going to take the symmetric group S3 which contains the six permutations And let's take the subgroup generated by the permutation the 3-cycle 1 2 3 That actually coincides with the alternating group A3 The even permutations the identity the 3-cycle 1 2 3 and it's inverse 1 3 2 So let's consider those cosets there. So in this situation What we can look at is we have the identity times H Well, if you take the identity like we said before this is just going to be back the original subgroup H And so this will contain 1 the identity 1 2 3 the 3-cycle and it's inverse 1 3 2 So the coset represented by the identity is always just the original subgroup nothing big going on there Um notice if we take the 3-cycle 1 2 3 What's going to happen here is you're going to get 1 2 3 times the identity you're going to go 1 2 3 times 1 2 3 And then you're going to get 1 2 3 Times 1 3 2 so we have to simplify each of those Well, if you take if you take 1 2 3 times the identity you're just going to go 1 2 3 Oh, we have that element there If you take 1 2 3 times 1 2 3 you get 1 2 3 squared, which is actually 1 3 2 And then lastly if you take 1 2 3 times it's inverse 1 3 2 you're going to get the identity So although it got put in a different order when you take 1 2 3 h You're going to get all of the elements in h back So ironically multiplying by this permutation permutes the elements of h, but it's still the same It's still all the same elements. So we're going to notice here that And I don't need the color coding here anymore That the coset 1 2 3 h is just h It's just the subgroup h right here and we're going to see a very similar thing if we do 1 3 2 h You're going to get something similar. You're going to get 1 3 2 times 1 Which of course is 1 3 2 you're going to get 1 3 2 times 1 2 3 Which as those are inverses you get the identity and then lastly you're going to get 1 3 2 Times 1 3 2 which is 1 3 2 squared, which is actually it's inverse 1 2 3 And so that's that coset and so by similar reasoning We see that You're going to get 1 3 2 times h all right, and so The the these three cosets 1 h 1 2 3 h and 1 3 2 h all produce the same coset Which is just the original subgroup h right there. Well, is something different going to happen What if we take 1 2 times h in that situation? We're going to end up with 1 2 times the identity which is 1 2 Then the next one we're going to get is 1 2 times 1 2 3 And so think about what happens there 1 goes to 2 2 goes to 1. So 1 is going to be fixed 2 goes to 3. There's no other 3 So 2 is going to go to 3 and then 3 goes to 1 which 1 goes to 2 So we end up with the 2 cycle 2 3 And then the last possibilities we're going to take 1 2 times 1 3 2 And if you go through the product this time 1 goes to 3 So 1's going to go to 3 Next we're going to get that 3 goes to 2 and 2 goes to 1 And then lastly 2 goes to 1 and 1 goes to 2. So 2 is going to be left fixed And we have a 2 cycle again the other 2 cycle Which one's left? What did I say it was 1 goes to 3 and 3 goes to 2? I'm sorry 3 goes to 1 and so what you see here is that the That the Coset generated by 1 2 represented by 1 2 gives you all the 2 cycles 1 2 2 3 and 1 3 And I want you to convince yourself, you know pause the video if you need to if you take the the coset from The coset represented by 2 3 you get the exact same set and if you take the coset 1 3 It'll also give you the exact same set. So There turns out to be only two cosets There's the coset h and the coset 1 2 h although you have some variety on the representatives you could choose Now, I also want you to convince yourself that each and every one of these cosets actually is equal to its right coset So if we put the representative on the other side, we would get The exact same coset that we had before So this is another example where the left and right right cosets actually agree with each other and That's kind of interesting, but again, I want you to convince yourself why that is I'm going to look at just h 1 2 for example if you take 1 times h 1 2 Notice, of course, that'll just give you 1 2 again if you take 2 3 times 1 2 You're going to get 1 goes to 2 2 goes to 3 and then 3 goes What did I say 1 goes to 2 2 goes to 3? Great Then the next one. Oh, I'm sorry. I know what I'm doing wrong here. Something felt fishy I need to not take things from the coset. I need to take things from the subgroup right here So I need to take 1 2 3 times 1 2 because let's get into three cycles like that's not right If you didn't do this one, this is the correct one 1 goes to 2 2 goes to 3 great Then 3 goes to 1 and then 2 goes to 1 1 goes to 2 so you get You get the 2 cycle 1 3 that you see right there and then lastly if we do 1 3 2 times 1 2 You're going to see that 1 goes to 2 2 goes to 1 so 1 is fixed 2 goes to 1 1 goes to 3 and then 3 goes to 2 which gives you the 2 the 2 3 2 3 right there that 2 cycle So even though you move things the other side you do end up getting the same coset the left and right coset Now this is not always the case. Let me show you one final example here in this video This time we're going to stick with the same group So g is still going to be the symmetric group s 3 but this time take the cyclic subgroup generated by 1 2 What happens in this situation? Well, I'm going to show you that if you take the identity times k This is going to equal of course just k itself Which is the group 1 and 1 2 And I also want you to convince yourself that if you take 1 2 times k that's going to be the same thing All right Now admittedly in this situation if you take k 1 That's the same thing because the identity commutes with everything. It's central to the group. Therefore that's not going to change And it's also true that if you take k times 1 2 That's going to give you that'll give you the same coset because these are all just equal to k in that situation So it's like keep on showing all these cosets that are the same left and right but fine fine here We are here. We are here's the here's the point now. We want to get to what happens when we take 1 2 3 times k All right, if you take 1 2 3 you're going to get 1 2 3 times the identity which is 1 2 3 and if you take 1 2 3 times 1 2. I think we already did this calculation Let's just do it again for the sake of practice 1 goes to 2 2 goes to 3 Great and then 3 goes to 1 great and then you see the 2 is fixed So you get the you get the 2 cycle 1 3 like so wonderful on the other hand And this is going to be the same thing as 1 3 which is k right there And let's let's finish up our list 1 3 2 times that by k Again, I'm going to kind of speed through the calculation. You're going to get 1 3 2 times the identity which is itself and then you're going to get 1 3 2 times 1 2 I want you to convince yourself. That's the 2 cycle 2 3 All right, so this this also is the coset the left coset 2 3 k these are the left cosets Let's then set these side by side with the right cosets if this time we take If we take k times 1 2 3 what happens? Well, you're going to get the identity times 1 2 3 which is 1 2 3 But then the next thing we're supposed to take 1 2 Times 1 2 3 which this time you get 1 goes to 2 2 goes to 1. So 1 is fixed 2 goes to 3 and then 3 goes to 1 which goes back to 2. So we actually end up with the 2 cycle 2 3 Which this is the same thing as the right coset k times 2 3. So notice the disagreement right here 1 2 3 k is not the same thing as k 1 2 3 because 1 2 3 k contains the 2 cycle 1 3, but the right coset contains the 2 cycle 2 3 So those things actually disagree with each other like I told you they would 1 2 3 k does not equal k times 1 2 3 All right, we also saw that if you take the left coset represented by 1 3 It'll contain 1 2 3 and 1 3, but if you take the right coset represented by 2 3 you're going to get Um, oh, I'm sorry. I'm looking at the wrong one. Here's the here's 2 3 k there If you take the coset the left coset for 2 3 you're going to have 2 3 and 1 3 2 But the right coset will contain 2 3, but it will contain 1 2 3 right these three cycles are different That's not the same coset and so what we've now seen is that the Left coset for 2 3 Is not the same thing as the right coset for 2 3 and we're going to see something similar happening right here if you take If you take the coset the right coset for 1 3 2 you end up with the elements 1 3 2 and the 2 cycle 1 3 I'll let you calculate that to convince yourself of that and so this is the right coset for 1 3 2 and the Right coset for 1 3 And you see some disagreement going on here the left coset for 1 3 2 contains itself But it also contains 2 3 but the right coset contains 1 3 So you see that 1 3 2 are excuse me 1 3 2 k does not equal k 1 3 2 so those cosets disagree with each other left and right cosets And then finally we also saw that the right coset for 1 3 contains the 3 cycle 1 3 2, but the left coset contains Excuse me contains 1 2 3, which would be this one right here And so then summarizing we see that 1 3 k does not equal k times 1 3 So in the case of the symmetric group we see that for this cosets associated to the cyclic subgroup generated by 1 2 The left and right cosets do not necessarily agree with each other That's something we have to pay attention to that the left and right cosets in general do not agree This is very common for non-Abelian groups. They necessarily agree for Abelian groups, but for non-Abelian groups They can disagree with each other. Do they automatically disagree? No We saw that the cosets for the alternating group actually were the same left and right But in general we cannot anticipate that they're going to equal each other