 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. In lecture 19, like I promised in lecture 18, we're going to continue our discussion of cosets. And specifically, I want to introduce right now the idea of the index of a subgroup. So imagine that G is a group and H is some subgroup of that. We define the index of H inside of G, which will be denoted as brackets G colon H brackets. Sometimes people use parentheses in this regard. That's a common notation used here. We'll use brackets though. So bracket G colon H, that's going to be the index and that's going to be the number of left cosets inside of G. So in other words, the index of H inside of G is equal to the number of left cosets. Now we saw in the previous video that this is equal to the number of right cosets as well. And so the fact that we define index between left cosets is really just a matter of preference. It doesn't matter. So if you hear me say, oh, the index is the number of cosets. The fact that I'm distinguished between left cosets and right cosets is not much of a consequence here whatsoever. So let's look at some examples. Let's revisit some examples we saw previously. So recall that if H is the cyclic subgroup generated by 3 inside of Z6, we saw previously that the cosets looked like the following. 0 plus H was equal to H. We had that 1 plus H was a coset. We had 2 plus H was a coset. And we could be specific here. We have 0, 3, oops. We also had 1, 4, and we had 2, 5. We also saw that the coset 3H was the same as 0H, that 4H was the same thing as 1H, and that 5H was the same thing as 2H. And that's because 0, 3, 1, 4, and 2, 5 represent the same left cosets. This is an abelian group, the right cosets, it's the same. And so we see here that there are exactly 3 cosets. And so this would tell us that the index of Z6 with respect to the cyclic subgroup generated by 3, this is going to equal 3. The index of this subgroup is 3. Let's take a look at the symmetric group for example. We saw some examples there where if you take H to be the alternating subgroup, then H, in that situation, we have the identity 1, 2, 3, and 1, 3, 2. This was identical to the cosets 1H, which we usually don't write as the identity, like when you see something like 0 plus 8, usually it's just called H itself. So the coset represented by the identity, we often just write as the, we don't put 1H or anything like that, you just write H. And likewise, this is the same coset as 1, 2, 3, H, and 1, 3, 2, H. And then we had the coset 1, 2, H. That was actually the set of all transpositions 1, 2, 3, 1, 3, and then 2, 3. Like so. This coincided with also the cosets 1, 3, H and 2, 3, H. And so we see that for this subgroup H, there's only 2 cosets. And so the symmetric group, the index in the symmetric group, the index of H is going to be 2. So there's 2 cosets. And then this group H3 here for A3, it was actually an example where the left cosets and the right cosets are the same. So these are identical, but you still had 2 cosets either way. If we take K for example, take K to be the cyclic subgroup generated by 1, 2. You have 1 and 1, 2. This was of course the same thing as the coset 1, 2. Some other cosets we saw. If you take the coset for 1, 2, 3, this contained 1, 2, 3. And then who was the other element here? Have to pay attention to that. So you take 1, 2, 3 times 1, 2. 1 goes to 2, 2 goes to 3. So this is going to contain the element 1, 3. And thus is the same coset as 1, 3. 1 goes to 2, 2 goes to 3. Yes, that's what it's going to be. And then the other one, if you take 1, 3, 2. This will contain, since it's a partition, I know what the other elements are going to be. You can take 1, 3, 2. And you're also going to have 2, 3. Now with this group K, the left cosets and the right cosets did not agree with each other necessarily. But the number of cosets is going to be constant. In this situation, there are 3 cosets. And so the index of K inside of S3 is equal to 3. And so that's what we mean by index. We just count the number of cosets. Whether left cosets or right cosets doesn't matter. The number is going to be the same. But the index is the number of cosets in a group with respect to some fixed subgroup.