 So let's continue our discussion about group isomorphisms. So let's suppose I have two groups, G and H, and I want to construct an isomorphism. Now, I don't know about you, but I like to make sure that what we're trying to do is actually even possible before we take the first step. And so one of the very simple things we can check is that because an isomorphism helps us identify when two groups that appear different are essentially the same group, we ought to check a couple of important things. The easiest thing to check is to make sure that the two groups are the same size. If they're not the same size, there is no hope that the two groups will be the same, and so there's no point in even beginning to look for an isomorphism. Now, after we've run this check, we might consider the three properties of isomorphisms that we found out last time. First, an isomorphism has to map identity elements to identity elements. Second, isomorphisms map inverses to inverses. And finally, isomorphisms map generators of subgroups to generators of subgroups. And here's a useful idea in mathematics that we've referred to before. Anything you can do once, you can do as many times as you need to. A subgroup is a group. It lives in a larger group, which is why it's called a subgroup, but it's a group. And if I have an isomorphism, that means I have to be able to map groups of the same size to each other. So what I might try to do is to look for two generators that produce subgroups of the same size and map the generator of one subgroup to the generator of the other subgroup. For example, let's say I have two groups, G and H, and I may have the following multiplication table for G and the following multiplication table for H, but my group elements are different. G has elements E, the identity A and B. Meanwhile, H has group elements I, the identity Delta and Sigma. Well, we can try and construct an isomorphism. So I only have three elements to map in G, E, A and B. So what's F of E? Well, the first thing we know is that isomorphisms map identity elements to identity elements so that F of E must be I. Now let's consider F of A. In group G, A generates a subgroup, A, B, E. Well, actually it's the entire group A is a generator, and so that means I want to look for an element of H that also produces a subgroup with three elements. And I see that the element Delta also generates a subgroup Delta, Sigma, I. So A generates a subgroup with all three elements. Delta in H generates a subgroup with three elements. So it seems reasonable to map F of A to Delta. And then finally, since an isomorphism must be a one to one function, F of B can't be I, can't be Delta, and because an isomorphism has to be onto, F of B has to be whatever's left over. In this case, F of B must be Sigma. Now, here's an important question to ask. Is this the only isomorphism between G and H? Or is it possible I could have made F of A something else, F of B something else? And the answer to that question is, put it in the water, see if it swims like a duck. Check it out because that's the easiest way of answering that question. Let's take a different set of groups, groups G and H, and here I have the multiplication tables as shown. G has four elements, A, B, C, and the identity. H has four elements, A, B, C, and the identity. And so to see if I can find an isomorphism between the two groups, let's generate some subgroups of G and H. So I'll generate the subgroup of G that is generated by A, and A times A takes us back to the identity. So this subgroup only consists of the elements A and E. It is a subgroup with two elements. So I'll look in H for a subgroup with two elements. And so I'll start with the element A and look at the subgroup generated by A. And if I look at the powers of A, that gives me A, B, C, and then finally the identity. So this is a subgroup of four or four, not what I want. I'll take a look at the element B in H. The subgroup generated by B has the elements B, A, C, and E, also a subgroup of four or four. And then third times the charm, if I take a look at the element C and look at what its subgroups are, those elements are going to be C, B, A, and E. Again, a subgroup of order four, maybe third time is not the charm, it appears that H does not have any subgroups of order two. And that is a very important result because isomorphisms have to map generators of subgroups to generators of subgroups of the same size. And so that means that A in G has nothing it can map to because nothing in H produces a subgroup of order two. And that means that there is no isomorphism possible. These two groups G and H are fundamentally different. They cannot be considered two different names for the same group.