 Let's continue our discussion about group isomorphisms and consider some isomorphism f that goes from some group g to some group h. The existence of an isomorphism between the two groups means that some of the properties of the two groups should be obvious, which is another way of saying that these are things you should be able to prove without difficulty and you should take the time to prove them. So some of these obvious properties include the following. First of all, g and h have to have the same number of elements. Next, there has to be an inverse mapping from h back to g, and this inverse mapping is also an isomorphism. Now again, part of the rationale behind setting abstract algebra is to develop your mathematical sophistication to develop your ability to think like a mathematician. And so here's another good habit you want to develop as a mathematician. When you get a shiny new object to play with, like isomorphisms, the question that you want to ask is, well, how do these things behave? What's the type of things that they do? And in this particular case, one way we can pursue this is the following. Suppose I begin with an isomorphism f going from group g to group h, and suppose I know that element a in g has a certain property. I don't know what that property is, but it's some group property that we can identify, and the question at hand is what can we say about f of a? And at this point, we introduced another useful thing to do as a mathematician is to recognize when a problem is too hard. So we'll start with an easier problem. What about f of e, where e is a specific element, the identity? And so let's talk about ducks. If you want to know if something is a duck, one of the things you can try to do is put it in the lake and see if it swims. If it sinks to the bottom, it's probably not a duck. So if we want to know something about f of e, let's see how it acts. Let's put it in the water and see if it sinks. So, well, we know that f is an isomorphism, so we know that f preserves the group operation. So f of e times a is going to be f of e plus f of a, where, again, the times indicates the group operation in g, and the plus indicates the group operation in h. Now, remember that e is the identity in the group g, but e times a has to be a, which means that I can replace e times a with a, and also plus is the group operation in h. And so what I have is f of a is something plus f of a. Well, that looks an awful lot like the identity. And remember, the identity of a group is unique, so if something walks like the identity, quacks like the identity, it must be the identity. And so that tells us that f of e is going to be the identity in h. And this leads to an important result, which is that an isomorphism must map the identity element of one group to the identity element of the other group. Well, there's another special type of group element, which is the inverse of something. What can we say about f of a inverse? And in particular, mathematicians are interested in relationships. How is f of a inverse related to f of a? And it's helpful to keep in mind what we know about a inverse. And the key idea here is that a times a inverse has to give us the identity element. And again, products of this type are part of how we define an isomorphism because isomorphisms preserve the group operation. So I know that f of a times a inverse must be f of a plus f of a inverse. And again, the times is the group operation in g, the plus is the group operation in h. Now over on the left-hand side, I know a times a inverse is the identity element, so I'll replace that. And over on the right-hand side, I have the product of two things equals the identity, which means that f of a inverse must be the inverse of f of a. And so this gives us a second useful result. Isomorphisms map inverses to inverses. Hey, that was a lot of fun and very useful, so we've identified two special group elements, the identity, and the inverse. Is there any other group element that's special? And one of those special elements is the generator of a group or possibly a subgroup. So again, a quick reminder, if I have some element of the group, then if the group is finite, there's some least value, p, for which a to the power p is going to be the identity, and also the terms in the sequence a, a squared, a cubed, and so on, up to a to the power p equals the identity. These are going to be distinct, and they will form a subgroup. And so that means that a is another special element of our group, so we might ask the question, what do we know about f of a? So again, let's take an isomorphism, and suppose I have some element that generates a subgroup, and because it preserves products, then f of a to the power n is f of a to the power n for all n. And since f is one to one, then because a, a squared, a cubed, and so on, up to a p are distinct, then f of a, f of a squared, and so on, up to f of a to the p, must also all be distinct. But by the proceeding, this sequence is just the powers of f of a. And this sequence is going to form a subgroup of h. You should prove this, it will be the subgroup that's generated by f of a. And this gives us another very useful result, which is that an isomorphism will map the generator of a subgroup to the corresponding generator of a subgroup. All right, so let's open up duck season. If I have an isomorphism, then we've determined three useful things about it. First off, f has to map the identity to the identity. Second, f has to map inverses to inverses. And third, f has to map the generator of a subgroup to the generator of a subgroup. And if I have a one to one onto function that does not do these three things, then I can't have an isomorphism. What I have is not a duck. And what this suggests is that this is a really good way of determining whether or not the thing that we have in front of us is an isomorphism. If it fails any of these things, then it's not an isomorphism. So we have a sure-fire way of identifying when something is not an isomorphism, which is a useful thing to have, although we may want to do a little bit more and ask the next obvious question, how do you construct an isomorphism in the first place? And we'll take a look at that next.