 One of the reasons mathematics is important is because we can take things that look very different and identify that they're really different versions of the same thing. And this falls under the category of what's called a group isomorphism. So let's consider two groups. We're going to take a group G, where the identity is E and the elements are A and B, and a second group H, with identity I, and elements delta and sigma. Now earlier we produced the only possible multiplication table for a group with three elements, which looks like that, and because this is the only possible multiplication table for a group with three elements, when we try to find the multiplication table for H, we find the multiplication table looks like this. And now let's do a little bit of an observation on our two groups. We see that group G is abelian A times A times B, or whatever it is. The products are symmetric because your multiplication table is symmetric along the diagonal. And if I look at the group H, I see that the H is also an abelian group. Both A and B have order three. A to the second is B, A to the third works out to be E. So both A and B have order three in group G. And if I look at the corresponding elements in group H, delta and sigma likewise have order three. Delta times delta is sigma, delta times delta times delta again, that's sigma times delta is going to be the identity. So both of these have order three as well. And the other thing that's worth noting, A and B are inverses, A times B gives you the identity, B times A gives you the identity, and likewise in our other group, delta and sigma are inverses, delta times sigma gives you the identity, sigma times delta gives you the identity. And we have two groups, well, they both quack like a duck. They both swim like a duck and they both walk like a duck. They seem to be the same group and the only real difference between the two is that the elements have different names. So because this seems to be the only actual difference between the two groups, maybe we can find a renaming function that's going to take the elements of G, E, A and B, and rename them, relabel them, I, delta and sigma. And one way we might do that is our renaming function might be the simplest one possible. The identity of G goes to I, the element A goes to delta, the element B goes to sigma. And because every element of G is assigned at most one value of H, we have a function from G to H. In addition, every element of H comes from some function value of G and so we have an onto function. And also, this is a one-to-one function. No value of H comes from more than one value of G. So we have a function that is both onto and one-to-one and maybe that's characteristic of when two groups have R in some sense the same. So the natural question asked is, is this enough? So in other words, is it enough for a renaming function to be one-to-one and onto? So let's think about that. Well, here's an obligatory joke. How many legs will a dog have if you call a tail a leg? Well, four. Because calling a tail a leg doesn't make it one. And what's important to understand is whatever our renaming function is, it can't just rename the elements. It has to preserve the properties of the elements. So we also found the multiplication table for groups with four elements, for some of the groups with four elements. We had two groups with four elements, the identity E, and named those elements A, B, and C. And one multiplication table we found by getting on board the bus A times A gives us the identity. And the other multiplication table we found by getting on board the bus A times A equals B, and that gave us this. And we can compare our two groups again. And G is abelian, so is H. G has three order two elements. A times A gives us E, B times B gives us E, C times C gives us E. So A, B, and C are all order two elements. H, on the other hand, has no order two elements. A times A, not the identity. B times B, not the identity. C times C, not the identity. There's zero order two elements in H. And that suggests that G and H seem to be different groups. That no renaming will make one of them into the other, because those order two elements in G don't have anything in H that has the same property. So, going back to this idea, our renaming function has to preserve the properties of the elements. Well, the key property of any element of a group is the product. So if I have a one-to-one-on-two function from G to H, I want those products preserved. So A is going to be assigned to F of A, whatever that is. B is going to be assigned to F of B. And I want A times B to be assigned to F of A times F of B. Whatever this product is, it's got to be assigned to the product of these two elements. Well, because the function works on the group elements, A times B has to be assigned to the function value of A times B. And what that means is that this value here and this product have to be the same thing. So I want to make sure that F of A times B has to be the same as F of A times F of B. And this has to be true for all elements A and B in G. Now, there's one possible point of confusion here. It's important to recognize that this time symbol on the left-hand side involves a multiplication of elements in G. So this time symbol corresponds to the group operation in G. Meanwhile, this time symbol, because these, F of A and F of B, because those are elements of H, this time symbol actually refers to the group operation in H. And what that means is that there could be some confusion here. And so we'll try to use different symbols for these different operations. So we might use this circle times to indicate the group operation in G. So I'll replace that here. And then I might use a circle plus to indicate the group operation in H. So this is a group operation in H. So is this, I'll replace those and write my statement that way. And this allows us to introduce a very important concept, which is that of isomorphism. So if I have two groups, G and H, with operation circle times circle plus respectively, I can take a function from G to H. I'll call it an isomorphism if the following things hold. First, F has to be a one-to-one function. F also has to be onto. And F has to preserve these operations. So F of a times b in G has to be the same as F of a times F of b in H. And this has to be true for all elements of G. And additionally, if I can find such an isomorphism, if I have two groups and if I can find an isomorphism, then we say that G and H are isomorphic and we write G isomorphic to H using this particular notation here. And what that means is that in some sense, these two groups are the same group. The only difference between these two groups is how we name the elements. If I can find an isomorphism, these two groups are not essentially different.