 There are many important transitions towards becoming what's called mathematically sophisticated, but one of the more important is the introduction to abstract algebra, and abstract algebra is centered around this notion of a thing called a group, which leads us to the following party game. How can you tell when you're dealing with a mathematician? Well, there's a number of ways that you can determine this, but here's an easy one. Ask them to complete the following sentence. There's a 1.2 of people over there, and a mathematician is going to use words like, well, there's a crowd of people over there, there's a bunch of people over there, there's a lot of people over there, there's a collection of people over there, they might even say there's a set of people over there, and they'll use many, many other words, but the one word that you'll have to really fight to get them to say is they'll hesitate to say that there is a group of people over there, and mathematicians really hesitate when using the word group in any other context, and that's because group has a very specific meaning in mathematics, and that meaning comes as follows. Let G be a set, and we'll define a binary operation star, and this binary operation has the following properties. First of all, if I take two elements of the set A and B, then A star B is in G. So remember, this is a binary operation. I'm going to take these two things, A and B, and I'm going to produce something new, and the result of that operation is in G. We refer to this property as closure, and say that G is closed under star. Another property of star is that if I take a star B star C, where I evaluate B star C first, and then star that with A, I get the same result as doing A star B first, following that up by star C. Now we have a name for this property, an ordinary arithmetic. We call this the property of associativity, and so in a fit of creativity, we'll also say that star is associative. One more property that we need for a group. There's an element E for which A star E is the same as E star A is the same thing as what we started with for all A in G. Here's an important idea. It's the same element is going to make this true for all A, and we say that E is the identity element. And then finally, if I take any element of G, there's some inverse, A inverse, so that A star A inverse is the same as A inverse A star A is going to give us back the identity element. And if I have all of these properties in my set with operation, then I have a group. One important feature of a group is that commutativity doesn't have to exist with the operation star. A star E and E star A, those have to be the same thing, and A star A inverse A inverse A star, those have to be the same thing, but in general, the operator star does not have to be commutative, do not assume A star B and B star A are equal. The most common mistake improves an abstract algebra and really in higher mathematics. The most common mistake is assuming commutativity when you don't actually have it. So here's an important thing to watch out for. Now let's take a look at some examples. We want to prove or disprove the set of natural numbers is a group. And so what we might do is we might put down our definition of a group and go through our checklist of requirements. So here's our definition of a group and let's see. Well, the set of natural numbers, so G is a set, so there's our requirement. The set of natural numbers is actually a set, so right here on the package. We need to define an operation star. Well, we don't actually have an operation defined. We have a set of natural numbers, but we don't have any defined operation, so that means we do not have a group. Now in your development of mathematical sophistication, here's a good habit to develop as a mathematician. A mathematician likes to turn one problem, one question, into many problems. So here, prove or disprove the set of natural numbers is a group. Done, this is not a group, but the natural next question that a mathematician might ask is, well, how can I make it one? If I don't have a group, how can I fix it? What can I do to make it into a group? So let's make one problem into many problems. The set of natural numbers is a group, prove or disprove. Well, we already disproved it, but then if it's not a group, how can I make it into a group? So again, we'll put down our definition. And the problem that we ran into is we didn't have a defined operation. So we need to define an operation star. So let's define a star b to b. Well, it's a binary operation, so I need to figure out what I'm going to produce when I start with a and b. I'm going to take a, I'm going to take b, I'm going to do something, and I'm going to produce something new. So thinking, thinking, thinking, how about something really simple, like define a star b to be a plus b? Well, now let's check. We've defined our operation, and so now we just have to check its properties. So first property, if a, b is in G, then a star b is in G. If I have two natural numbers, well, so is the sum of those two. That's also a natural number. Let's check our associativity, a star b star c, a star b star c, a plus b plus c, a plus b plus c. We'll verify, and if a, b, and c are all natural numbers, we know that the natural numbers under addition are in fact associative, so that also works. And let's see, we need to, oh, here's a problem, we need to have an identity element, a plus e has to give us what we started with. Well, because we're dealing with a set of natural numbers, this is not going to be true for the set of natural numbers. The identity does not exist. Now we can fix this, how can you make it one? We can fix this by changing our set from the natural numbers to the whole numbers. So let's go ahead and make that swap. So I'm going to swap out and deal with the whole numbers. And again, we'll define a star b to be a plus b, and because we've changed our set, we do have to go through all of our requirements once again because it's possible we've introduced something new. So let's check it out. We have closure if the sum of two whole numbers is a whole number. We have associativity. We have the identity element because the number zero is our identity. However, we don't have an inverse element. I can't find two whole numbers that add to the identity. If I have the whole number five, there's no whole number that I can add to five to get the identity zero. We don't have the inverse element, which means we have to change our set again. So let's expand our set to the integer. And so let's check to see if the integers form a group. So again, we'll continue to define a star b to be a plus b. And once again, we do have to go through all of our properties just to make sure we haven't lost anything in the process. We have closure. The sum of two integers is an integer. We have associativity. We have the identity element zero again. And because we're dealing with the integers, we do have the inverse. Given any integer, I can find another integer that when I add the two, I get zero. And so now the set of integers with the integer addition does form a group.