 Let's take a look at the identity of a group in somewhat more detail. So remember that in a group the identity e is something that made a star e the same as e star a equal to a itself. When I apply the operation times e, I get the thing that I started with. Now here's another useful habit to develop your mathematical sophistication is to wonder about the words that aren't spoken. When we define the identity for a group, we said it's an element that made this true. And the question might be, might there be a different identity element? Might there be something else which holds for all a? So to answer that, let's introduce another quote by one of the founding figures of American mathematics, Benjamin Perce, down at Harvard University. And he's a founding figure of American mathematics. This, by the way, is not a misspelling. His name is actually spelled P-E-I-R-C-E, and Perce is famous for many things, but one of his quotes is, mathematics is the science of necessary consequences. And what he means by that is, well, in mathematics you can get on a bus and see where it takes you. And in mathematics the bus is something you know, or at least you assume, to be true, and the journey is produced by the logical deductions. And mathematics is the things that we produce on that journey. So we start with something we believe to be true, or we agree to be true, and we see where it takes us. So, let's take a look at our identity crisis. Suppose I have E and E prime, and suppose there are two identity elements for some group G. So here's the bus that I'm going to get on. E and E prime are two identity elements for some group G, and we want to know how they compare. Now, a useful idea here is when you're comparing two things, the first key question you want to answer is, are they equal, or are they different? So we'll get on the bus and see where it takes us. So in this case, the bus that we're going to get onto is labeled with the one thing we know to be true. The one thing we're starting with, E, is the identity element. So, I know that E times A is equal to A for any element of the group. But if I know that E prime is also an identity element, then I know E prime star A has to be A as well. And, well, both of these are equal to A. Both of these are the same as A, which means that any time I see A, I can replace it with the other one. And so these two, E star A, E prime star A, have to be equal to each other. Now, because I'm dealing with a group, because I'm dealing with a group, I know that A has an inverse, A inverse. So I can multiply by it on the right. Here's an important qualifier. Remember, groups can't in general be trusted to be commutative. If I multiply on the left by A inverse, I can't switch things around to get the A and the A inverse next to each other. So I have to multiply on the right by A inverse. And if I do that, because I know I'm dealing with a group, I know I have associativity. So I can group, combine these last two elements here. And now I have A times A inverse, A times A inverse. And because it's a group, I know that that is going to be the identity element E. And because E is the identity element, I know that whenever I multiply by E, I get whatever I started with. So here on the left-hand side, I have E times E. So that's just going to be E by itself. Over on the right-hand side, I have E prime times E. And because E is the identity element, then this product is just going to be E prime. And I found something out. If I have two identity elements, E and E prime, they actually have to be the same thing. And so this proves an important theorem, which is that the identity element of a group is unique. More importantly, this particular structure of proof is a very common one in abstract algebra. Suppose I have two things. If I have two things that have the same property, I want to know something about how they relate. And so what I'll do is I'll use their property and I'll work through something and then I may find some relationship between them. In this particular case, it's that the two identity elements have to be the same thing. In other cases, there may be a different relationship between the two things that have the same property. But in general, we'll start out with what we know about them and work our way through our logical deductions.