 So now that we understand symmetry operations and the symmetry elements of a molecule, we're almost ready to understand the statement, which we'll really consider in the next video actually. The statement that the symmetry elements of a molecule form a group. But before we can understand what that statement really means, we need to understand what is the thing I just called a group in the mathematical sense. This general idea of group theory is the mathematics of these objects called groups. And it turns out a group, you're somewhat familiar with them by experience already, even if you've not heard that name before. A group consists of two things, it consists of a set of elements, so a set like you're familiar with from introductory math classes, and an operation that tells you how to combine two elements of that set. So to help make this a little more concrete and understandable, we'll use some examples. The main example we'll use to understand the properties of a group is the integers. So I'll let the integers 0, 1, 2, 3, and so on, and also the negative values, negative 1, negative 2, and so on. So that full set of integers is the set that we'll be thinking of, the operation with which we'll combine those integers. We can combine the integers in various ways, we can add them, subtract them, divide them. So the operation we'll consider is addition. So as we talk about the properties of a group, we'll use adding of integers to understand what those properties mean. So not everything is a group, a group is a set in an operation that has the following four properties. So there's four of them. The first one is that the operation needs to be associative in the same sense that in elementary school when you learned about the associative law and the commutative law and the distributive law, associativity means the same thing here. Whatever my operation is, so again, maybe my operation is addition or multiplication or some other operation. If I have some element of a set and I want to combine it with another element of the set, I don't want to write plus or times here because these are not rules specific to adding or multiplying. For my operation, I'm just going to use a circle. So a circled with b, a added or multiplied or whatever with b, a combined with this operation with b. If I do a and b and then combine it with c, combine the result with c, that's going to work out to be the same as a combined with the result of b and c. So that's the associative property like you're used to. As our example over here, it's certainly true that if I just take some of these integers, if I take 1 plus 2 and add that to 3, that's going to be the same as if I take 1 plus the result of 2 plus 3. You're certainly used to addition being associative. You use that almost every time you do any arithmetic or algebra. It doesn't matter in what order you add these numbers together, you get the same result. So addition is associative. Addition of integers in particular is also associative. So that property is satisfied by addition. The second property that a group needs to have is there has to be an identity element in the set somewhere. So what that means is that if I take an element from the set, one of these numbers, for example, there must be some element, the identity element, that I can combine it with and get back the same number as I started with. So the identity is a particular element of the set that no matter what element I combine with it, I get back the same element I started with. So the identity element in addition is 0. If I take 2 plus 0, I get back 2. If I take 3 plus 0, I get back 3 and so on. So no matter what element of the set, if I add 0 to it, I get back the same element I started with, so 0 is the identity. A key point about the identity is there's only allowed to be one identity element. The same identity element, 0, has to be the element that you add to any element to get back itself. The third, and in some ways the most interesting property, the most characteristic property of this groups, is that the set needs to be closed under the operation. That one might begin to sound a little bit unfamiliar, but what that means is if I take any element from my set, I combine it with any other element from the set, I'm going to get a result. That result needs to be in the set. So in other words, what closed means is the set is closed under this operation. If I perform an operation between two elements, the results of that lands back in the set. It doesn't ever land outside of the set. I don't get a new element when I perform an operation between these elements of the set. So A operating with B must be in the set. That's what it means for the set to be closed under our operation. For the case of adding integers, that again is clearly true. If I take 1 plus 2, I get 3, and that's already in the set. No matter how big the integers I take, if I take 100 plus a million, I get 1,100, that's also an integer. So I can't find two integers that I can add together and I won't get an integer. So that set is closed under addition. And then the last of the four properties that we need to consider is that every element of the set needs to have an inverse. So watch carefully the difference between the identity and the inverse because it's easy to get these confused. There's one identity element that works when you combine it with any element. Inverse says, if there's an element of the set, there must be some inverse that I can combine it with. I'll write that as A inverse. And when I combine A with its inverse, what I get back is the identity. So if this is one of the elements in the set, this also must be in the set. But every element is allowed to have its own unique identity if it wants to. So how does that work for addition? Let's say I take the number five. What is the inverse of five? What number do I combine? Do I add to five to get back the identity? Identity was zero. So if I add five and negative five, I get back the identity. I get back zero. If I combine four with negative four, then I get back zero. But notice that the inverse of five is negative five. The inverse of four is negative four. Every element of the set is allowed to have its own individual inverse. There doesn't have to be a single inverse, like there does for the identity. So those are the four properties. If you have a set, you have an operation for how to combine elements of the set. And the set under the operation has these four properties. Then we say that that set and that operation form a group. So our example has shown us by validating each of these four properties that the integers under the operation of addition do form a mathematical group. So integers form a group under addition is the way we would phrase that. One important thing to notice is that a group is not just a set and involves both the set and the operation. So it's not correct for me to say the integers form a group, for example. I would have to say the integers form a group under addition. If I were to choose a different operation, let's say I were to take integers as my set and multiplication as my operation. Multiplication. So I can ask integers under the operation of multiplication does that or does it not form a group? Well, we would have to go through and check each of these properties. Starting, it might intuitively seem like they form a group. Multiplication is associative, it doesn't matter what order I multiply numbers in. There is an identity, what number can you multiply an integer by to get back the same number? That number is one. So the multiplicative identity is one. One is a number that's in the set. So there is an identity, multiplication is associative. Is the set closed? Can I take any two integers and multiply them together and get something that's not an integer? Nope, I can't. Any product of two integers is going to be another integer. So far so good. The set is closed under multiplication. But it turns out there's not an inverse. If I ask, what do I need to multiply five by to get the multiplicative inverse? What do I need to multiply five by to get one? I need to multiply it by one-fifth. The multiplicative inverse of five is one-fifth. But one-fifth is not an integer, that's not in the set. So because many elements of the integers, their inverse is not in the set of integers. It doesn't obey this fourth condition. So multiplication of integers does not form a group. So that illustrates this point that you can't say whether the integers are or are not a group. They are a group under addition, they're not a group under multiplication. A couple of other features to point out about groups. Many of the groups we can think of using ordinary mathematics are infinite sets like the integers. There's an infinite number of integers. And that seems reasonable because every time I combine two integers, I might get something larger. So I keep having to make the set larger when I combine the integers to make larger ones. But it turns out there are groups that are not infinitely large. So just as one final example, and we won't work through it in detail. But the numbers on a clock under the arithmetic of clock arithmetic form a group. So the numbers 1 through 12, 1 o'clock, 2 o'clock, 3 o'clock, up to 12 o'clock. And then adding those times together form a group, even though there's only 12 of them. So for example, that set is closed because if it's 10 o'clock now and I want to know what time it is four hours from now, it won't be 14 o'clock. If you're using a 12 hour clock, it'll be 2 o'clock. So once you go off the top of the 12 o'clock list, you circle back around to 1 o'clock, 2 o'clock. So that would be an example of a group with only a finite number of elements, only 12 hours on the clock that is closed. You can think about it hard enough to decide that it's also associative. There is an identity, there's also inverses. So there are finite groups and in fact, the elements of a group don't even have to be numbers like they are on a clock or in the integers. It's possible to have a group composed of elements that are not numbers at all. And in fact, that's what we'll do when we talk about molecules. The symmetry elements of those molecules are not numbers, but they happen to form a group when we combine them in the right way. So that's what we'll talk about next.