 All right, so let's see what we can understand about the symmetry elements of a molecule now that we understand a little bit about group theory. So just as a reminder, we've seen for molecules like H2O that we can find all their symmetry elements, find all the symmetry operations that leave them unchanged. And for water in particular, there's an identity element as always. There's a C2 rotation. If I rotate water by 180 degrees around that axis, it remains unchanged. There's two different reflection planes. There's one that's in the plane of the board that cuts every atom of the molecule in half. And then there's one that bisects the molecule as well. So we call those both vertical reflection planes because they contain the C2 axis. They're standing up straight vertically if the C2 axis is pointing upwards. So those are the only symmetry elements of a water molecule. So my claim has been that the symmetry elements of a molecule form a group, and in particular they form a group under this operation of composition. If we combine two symmetry elements by performing them back to back, then under that operation the symmetry elements form a group. So we can check whether that's true for this case of water, and the most difficult part of it to check is checking whether these operations, whether these symmetry elements are closed under composition. If I can combine, every time I combine two of these elements, I get another element that's already in the group. So to double check that that's true, we can make ourselves a brief little multiplication table. So we'll just go through all the cases of multiplying each of these symmetry elements or symmetry operations by another and seeing what we get. And let's go ahead and label sigma v being the one that bisects the molecule in half. In this direction, sigma v prime being the plane that contains all three atoms of the molecule. So many of the elements, there's 16 different products we need to evaluate. Many of them are fairly simple. The identity element in particular, by definition, if we do an identity followed by or preceded by any of these other elements as if we didn't do anything when we did the identity. So identity combined with identity just gives us identity. Identity combined with c2 gives us a c2 with sigma v and sigma v prime gives us those. And likewise for this column of the table, identity with c2, identity with sigma v, identity with sigma v prime. Some of the other combinations are simple to do without doing any complicated 3D visualization. C2 and a c2 rotate halfway around, rotate the rest of the way around, gives us the same as if we had done nothing. Likewise reflection plane sigma v followed by sigma v prime, I'm sorry, sigma v followed by sigma v gives us the identity reflection through the sigma v prime plane followed by reflection through the same plane is the same as if we had done nothing. So we really only have six of these that are challenging to figure out. And those are in fact the six that we had worked out using the cube visualization example in the previous video lecture. So there we saw that if we did a c2 followed by a sigma v, then that was the same as a sigma v prime. So a c2 followed by a sigma v prime was the same as a sigma v and a sigma v followed by a sigma v prime is the same as a c2. And likewise for the other half of this table, if we do those operations in the reverse order, in this case it doesn't matter what if we reverse the order. Sigma v followed by c2 is a sigma v prime, sigma v prime followed by c2 is a sigma v and sigma v prime followed by sigma v is a c2. So and again, those last six that we entered, that's not easy to tell immediately just by looking at the symmetry operations themselves. You have to sit down and either play with your hands or draw some pictures of a 3D object and imagine what happens to it as you reflect it or rotate it. But those were the reasons we worked those out as the examples in the previous lecture. So now that we've filled out this multiplication table, what have we learned? Well we've learned a few things. First of all, there's no entry in this table that isn't already one of the four symmetry elements. I didn't combine any of these two elements and then discover a c5 or discover an inversion or discover an s2 or something like that. The elements in this multiplication table were already in the collection of four. So in fact, the set is closed under composition. And that's actually not a surprise now that we think about it. The reason that these were the symmetry elements of the molecule is because that's all of the operations that leave the molecule looking unchanged. If I do any one of these, the molecule is unchanged. If I then do another one, the molecule is unchanged. And it's unchanged in a way that's described by one of the other symmetry elements. So it's actually not surprising that the set is closed under composition. We also need to have an identity. So the set is closed. Do we have an identity? Sure. In fact, we call it the identity. So every group of symmetry elements has this identity operation. Is there an inverse? If I do c2, is there something I can combine it with to get an identity? Sure. c2 followed by c2 is identity. Reflection followed by the same reflection as identity. A different reflection followed by the same reflection as identity. So there are inverses. Every element has an inverse. Even e has an inverse. Every element is its own inverse in this case. So there's an identity. And associativity, we won't spend any time trying to demonstrate, but associativity is also true. If I do one operation followed by another followed by a third, it doesn't matter whether I do the second and third. And then first followed by the second and third together, or the first and second followed by the third together. I can't shuffle the orders necessarily completely, but they are associative. So symmetry operations will always obey associativity. So therefore, those are the properties we need to validate to confirm that the symmetry elements form a group under composition. So all we've confirmed so far is that the symmetry elements of water, these particular four symmetry elements form a group. We haven't proved it for anything else. But it is, in fact, true. And hopefully after this example, not terribly surprising, that the symmetry elements of any molecule will, in fact, form a mathematical group. So the next step is to explore the properties of those groups a little bit further.