 Welcome back to our lecture series Math 4220 Abstract Algebra 1 for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. We're going to talk some more about groups today. We learned about them in the previous lecture, lecture 7. For lecture 8, I want to focus primarily on an aspect related to a group which is called a Cayley table named after mathematician Arthur Cayley here. A Cayley table of a group is a table, as the name might suggest, where the rows and the columns of the table are associated to elements of the group. Then when we look at the interior of the table, by taking the row associated to G and the column associated to H, the element in that row and column is the product of the elements G and H. We can actually use the Cayley table to represent the binary operation in play right here. I say product in sort of a general sense. I'm not saying the operation has to be a multiplication, but you can look up on the table what is the resultant given the two operands. As an example of this, let's take the group Z5, with addition, where we take the number 01234. We add them together by the rule of addition, mod 5, and the Cayley table would be what you see over here. We're going to have rows that are associated to, I guess I'm talking about the columns right now, we have columns associated to elements of the group, 01234. Then the rows are also associated to elements of the group, 01234. It's not required, but it's sort of best practice to write the rows and the columns in the same order. And we traditionally like to put the identity element first. Now, when you look at a group like Z5, as we're going to represent these numbers as integers 1 through 4, it makes sense to kind of put them in ascending order. For most groups, though, there's not some well understood notion of, you know, ordering right. And so there might not be an obvious listing, but we usually like to put the identity element first. And then we also typically, when we have a Cayley table, like to indicate the operation symbol itself, like this is a plus. And it's not required. But when we're working mod 5, it's useful to represent the operation. Because in addition to modular addition, there's like modular subtraction, modular multiplication, modular division. And the notion of a Cayley table actually makes sense for any binary operation. It does not have to be a group to have a Cayley table. We can make a Cayley table for any binary operation. But there are some properties we can notice when we look at the Cayley table here. So for example, if we pick the number 1 and we pick the number 3, if we come together, we find the number 4. What this means is 1 plus 3 is equal to 4 when we work mod 5. That's what we're trying to say right there. Clear it up a little bit. As another example, if we take the number 3 and we take the number 4, and then when we come together, we look that row, that column, we get the number 2. What this means to us is that 3 plus 4 is equal to 2 when we work mod 5. So we can read the binary operation off of the Cayley table by looking up the rows and we look up the columns here. Now the identity in a group can be identified as the unique element with a row or column which matches the group indicators perfectly. So let me kind of erase the screen right here. You'll notice that when you look at 0, like the row that involves 0 matches up with the indicator row perfectly. That's because it's the group identity. And likewise, if we look at the column associated to the identity and you look at the column of indicators, you'll see that the two things are identical. Every other row will be a rearrangement of the group elements. Every other row is a proper permutation. I mean, mainly this is the identity permutation, right? If you look at the column associated to 1, things got mixed around a little bit. 1, 2, 3, 4, 0. If you look at the column associated to 4, you're going to get 4, 0, 1, 2, 3. It's not the right order we started off with like right here. So when one looks at a Cayley table, you can actually identify the identity element because you're looking for the row that's just a carbon copy of the indicators on the Cayley table. So we can identify identities from the Cayley table. Also the inverse of an element, say g, can be found on the Cayley table by finding the column which contains the identity of the identity in that row. So for example, if you look at the number 1, we know who the identity is because the identity looks just like the row of indicators here. So because we know who the identity is, we can search this. And we find it. Here you are. So we find the identity in the table, and that then tells us that 4 is the inverse of 1 because 1 plus 4 is equal to 0, the identity. All right, go back up a little bit. If we were picked to say the number 3, we could see that if we search along, here's a 0. And then if you look up who gave me 0 as a 2, we see that the inverse of the number 3, mod 5, is 2 because 3 plus 2 is equal to 0. We can find the inverse of any elements. So like we said, 1's inverse is 4. Likewise, 4's inverse is 1. We saw that 3's inverse was 2. And likewise 2's inverse was 3. Who's the inverse of 0, the identity itself? Well, it's its own inverse. So we can find inverses by looking at the table right here. Another thing we can also see from the Cayley table is that this group is commutative. That is what we call an abelian group. This is what, the way we can see that is that this table, because honestly a Cayley table is just a matrix, right? This is just a 5 by 5 matrix, if we look at these numbers right here. Well, what happens when you take the transpose of this matrix? If you reflect this table across its main diagonal, you will see that it's a mirror image. You have 1, 2, we get 3 and 3 right here, 3 and 3 there, 4, 0, just looking at some of these examples. This table is symmetric. If we think of it as a matrix, this is a symmetric matrix. And a symmetric table, a symmetric table implies that you have a, we'll say, a commutative, a commutative operation. So summarize what we've done here. We can identify that the operation is commutative by seeing the Cayley table is symmetric. We can find the group, we can find the identity element by finding a row identical to the labels. And we, and if we have an identity, we can search for inverses by looking up the identity in the table itself. And we can also do all of the binary operations from the table. So one can very quickly see whether we have a group from a Cayley table with one exception. It turns out that associativity from the Cayley table is not an easy thing to identify because associativity involves three elements. A, B times C is equal to A times B, C. And so since it requires three elements, the table only talks about two. It's a little bit more difficult to see associativity from a Cayley table. So I'm not going to present a method of quickly identifying associativity from a Cayley table. In general, that's a hard argument to show that an operation is associative. Before we end this video though, let's look at another example. This time, let's look at the multiplicative group Z8 star with respect to multiplication mod 8. Remember how we constructed the group Z8 star? We only look for those numbers which are co-prime to 8. And so when you go from 0 to 7, that'll be the numbers 1, 3, 5, and 7. Notice very quickly we can identify that 1 was the identity if we didn't know that already because it matches up right here. We can also see that every element is its own inverse, right, because notice 1 times 1 is 1, 3 times 3, 4 times 4, and 7 times 7 is the identity. So every element is its own inverse, which is kind of curious. And also we can see that when we reflect across the main diagonal that this Cayley table is symmetric. So this tells us that this is an abelian group. Again, the associativity of multiplication is not determined by the table, but that's something we already know. So we can see some of this information from the table here. Something else I wanted to point out from this Cayley table here, I mean other than the basic facts that like 3 times 5 mod 8 is equal to 7, 3 times 5, which we know is 15, is equal to 7 mod 8. So we can read that from the table without having to do the calculation, assuming the table is small enough we can manage it that way. The other thing I want to mention is that in any row or column of a Cayley table, you'll notice that every element of the group appears once in only once. So like let's take the row associated to 5. We see a 5, we see a 7, we see a 1 and we see a 3. Every element shows up in that row in some order, could be scrambled up based upon some permutation here, but every element shows up once and only once. When you have a table where every row and column show, every element shows up once and only once, this is referred to as a Latin square. And you might have seen Latin squares before, because honestly every time you play the game Sudoku, you are experimenting with a Latin square. Sudoku, as a game right, has the rules that all the numbers 1 through 9 show up in every column exactly once and every row exactly once. I mean there's also these squares you draw when you have Sudoku, so it's a little bit more specific, but Sudoku is an example of a Latin square. Every column and every row in a Cayley table has the property that the element shows up once and only once. It will be there, but it will only be there once.