 Given two sets A and B, we can combine them in two ways. We can talk about the union, A union B, and we can also talk about the Cartesian product, A cross B. Now, if we have two groups, G and H, can we combine them? Yes, if we're careful. Well, let's think about that. G times an H plus B groups with operation times in G and plus in H. The important thing to realize is the binary operation times only works on elements of G, while plus only works on elements of H. And so the union, G union H, will contain elements for which times is undefined and will contain elements for which plus is undefined. And we'll have no idea how to handle expressions like A times X if A is an element of G and X is an element of H. Unions are useful. Well, OK, at least not for algebraic structures. On the other hand, remember that a Cartesian product takes elements from two sets and forms an ordered pair, and this suggests the following. Let G and H be groups with operations times and plus, respectively. The direct product of G and H, typically written G times H, is a set of ordered pairs, G H, where G is an element of our group G and H is an element of our group H, where we define the operation plus by, in other words, we operate on the elements of the ordered pair using the appropriate binary operation. And remember, there are only so many symbols. We have to recycle them. So this circle plus is a way that we can indicate that this is a new type of operation, but we shouldn't rely on having new symbols for every operation. And in fact, we'll adopt the convention that the binary operation is always understood and used juxtaposition. So AB plus CD will write as ACBD, where it's assumed we'll use the correct binary operation from each of the two different groups. Now, of course, the question you've got to ask is, do we actually have a group? And so we'll prove that if G and H are groups, the direct product is also a group. So our binary operation is closed and associative, and we'll leave the proof of closure and associativity to the viewer. To find an identity, we need some element X, Y in our Cartesian product, where for any G and H, we have GH plus XY still gives you GH, and also XY plus GH still gives you GH. Now, our definition of this binary operation tells us that the left-hand side here is GXHY or XGYH. And in order for these two Cartesian products to be equal, they have to be equal component-wise. So here on the left-hand side, we have GX must equal G and also XG must equal G. And this can happen if X is the identity element of G. And similarly, if we look at these second components, we see that HY must be H and likewise YH must be H. And so that tells us that Y must be the identity in H. And so the Cartesian product G cross H must have identity, e.g. EH, where EG and EH are the identities of G and H, respectively. Now, we have closure, associativity, and identity, and all we need at this point is an inverse. So if G and H are elements of our Cartesian product, then the inverse is also an element of our Cartesian product. And again, we'll leave the proof of the presence of the inverse to the viewer.