 So we've looked at dividing a group up you know and then getting a quotient group now that is something that we've put in our box keep it in your box we are going to revisit homomorphisms and isomorphisms by using that but before we do that whereas before we divide it let's now multiply so let's talk about the product group product group so what I want you to consider is just two are really arbitrary groups h and k they can be finite they can be infinite the sets can be totally different the group operations can be different and so they totally separate groups and I'm going to define the product of these two as such I'm just going to say just going to use the multiple universal multiplication sign there and if we just consider the set that is going to make up what I claim to be a group I'm claiming if I do this I just have to define something we decide how do we define the set that is going to make up this group is that I'm going to do this Cartesian product Cartesian product remember I take every element and I pair it of the one group and I pair it with the element of the other group so I'm going to have these two this this Cartesian pair those two such as h is an element of that group and k is an element of that I'm just considering this so remember I mean if the elements in the one are a b c and the elements here are x and y I'm going to have a and x and I'm going to have a and y you know and then I'm going to have b and x b and y b and x b and y and I'm going to have c and x and c and y so all these Cartesian pairs so they needn't be the same size as I said they don't you know they don't have to share any elements the set that makes up this one group and the set that makes up the other group they can be totally disjoint you know it doesn't matter what we do and I'm going to define you know I'm going to say that that is the set and now if that's the set I still have to make up a group operation so you know that's my set and now how do we define this group operation that we can have and we're going to define we're going to define it as such that if we take one element I'm just going to call it say h sub one and k sub one that was my one and I'm going to do the bind my binary operation which I'm claiming will form a group and I'm going to have another one say h sub two and k sub two my second Cartesian pair so I'm just defining what this is going to look like and I'm saying I'm going to define this as another Cartesian pair and it's going to be h one composed with h two those two and they both come from h and then k one and k two that is our human definition how we decide we are going to do that now we need to show that this product group this product is actually a group under this defined group operation now for me to show you that this is a group it must have all the properties of the group first one is closure and since you know this came from a group with its binary operation and we can actually make that this you know this has a different binary operation so it didn't have been that you know it doesn't have to share this binary operation but anyway that you know came from a group and there's closure there and there's closure there so we have closure associativity associativity well actually that's by our definition that we've chosen chosen here is going to is going to be very easy so imagine then I also have here I'm going to have h three and k three and if I do these two first you know I'm going to have the binary operation with those two comma the binary operation of those two so that gives me a new element and that gives me a new element and then the binary operation with that would just be the binary operation you know with these two then that one and if I did this first and then that one it doesn't matter I'm going to end up with I'm going to end up with this scenario and comma I was going to end up with that scenario if I did these two first and then that one or these two first and then that one I'm going to end up with exactly the same thing there's not going to be any problems there identity element well from the way that we've defined it that's very simple to to to show it is just going to be the identity element for my one group and the identity element of my other group and then I can show that if I if I were to plug those in in my in my original definition of my of how I do my group operation that I no matter if I do that first and then identity element or the identity element first and then this one I'm just going to end up with either this one or that one you can do all of the paperwork that's not an issue at all and then you can well imagine that so that's a unique one and then you can well imagine that the that the inverse element for each element inverse elements you know each of these each of these are going to have their own unique elements such as if I do the binary operation with the inverse I'm going to get the identity element or the other way around I'm going to get the identity element and that is very simple going to be the inverse and the inverse following from the way that I defined initially the group operation on my new group so that is all follows from how we define this group operation how we define it and from that follows very easily that it has all the properties so what we are just saying is that this whole thing here this whole thing here each of these is now a separate element even though they you know the two elements listed in there that that that Cartesian pair is just now a new element in a group you know and just because we we do it separately it doesn't doesn't matter it is now this is now a unique element inside of a group and then every element will have this in this group this group is going to have this inverse this whole group is going to have this unique identity element there's going to be associativity and of course there's going to be closure so very simple that is how we define a product group