 So now that I hope you understand group actions and have this deep intuitive understanding of the easy concept that really is going on here, let's just look at an example of a promised non-faithful group action. That is where we have any one single permutation of my set A being represented by more than one of the elements in G. So consider this one, the action of an arbitrary group G on an arbitrary set A is defined by the following. G.A equals A, so I'm not, you know, it's not one of those. So this A is really equal to that A, so this one is three, that one's also three. And I want to see do the group action, do the properties of group actions hold. And remember the first property I said there for all G1 and G2 element of G, that's my set. And then for all A elements in A, we have the fact that if we have G1 dot and I have G2 dot A, that that would be equal to G1, G2 acting on A. That that should be exactly, exactly the same thing. Now let's just have a look at this left-hand side. What does G, whatever the G is dot A mean? Well that by the way that we set up is just A. So on this side I'm just going to have A again, the one. But this G is just any G here, and that is just equal to A. So that is just A on this side. And on that side, well, there's closure, this is a group, there's closure. So this is just going to be some other G, let's call it G star dot A. And by my definition is, that's just equal to A. That's just equal to A. So the first property holds. And what we have for the second property is that we have E dot A must equal A. Well E is just one of the G's, and that must be A. So this is just A equals A. So that one also holds. And what you can think of here is that we are mapping all the elements in G. So if I have my G set, all of the elements in it, no matter which one they are, they just map to the identity permutation. The identity permutation. Because no matter which one I take, which one I take, it just leaves A, its action on A just leaves it in place. So one maps to one, two maps to two, three maps to three. So here I have a beautiful example of a non-faithful group action. So these permutations, the permutation has more than one element in the set. Actually, in this case, all the elements of my, in G is going to map to exactly the same permutation. And then think about this one as well, because that is actually how we set up, this is how we set up Cayley's table. It's also a group action. Because if I consider my group with binary operation zero, and that is the G, the set, and the binary operation, and instead of set A, I have this set. So it's the action of this set onto itself. So what we're saying here, my mapping is my, if I take the Cartesian product of the two, and you know, that's just exactly, I'm taking same elements, and that maps just back to G. That is the mapping that we have. In other words, what we're saying here, if I have G1 acting on G2 for all G1, G2 in G, that that would equal this. And I'm wanting to know, you know, do these group properties hold. So I'm saying here, if I have G1 dot G2 dot G3, that that would be equal to, you know, G1, and I have this binary operation G2 with G3. And, you know, with that hold, let's put it, you know, with that hold. And what we're saying here is if I take G and these two, I'm just going to end up with another G. So that's the, or the way that we've defined it here is that if I take this first G2 dot G3, so if I have a G dot a G, don't worry about the two and the three. I'm just stating elements would be equal to those same elements, the binary operation between them. So that would be G1 dot, the dot must be at the bottom, G2 dot binary operation G3. And on this side, I still have G1 binary operation G2 binary operation G3. And again, going according to this, is this going to be some element in G? So in our representation, that will represent this. And I'm saying again that the action of this onto that must just be the binary operation between those two. So this is going to be the binary operation of this and then equals G1, G2 with G3. And all we left with here is the associative property and these two are exactly the same. And the same argument is going to go for the identity element here. So the properties of group actions do hold. And what we have here is just the setup of, all we have here is just the setup of Cayley's table. So it's two lovely examples to help you understand group actions or just think about something in group actions that you might not have thought of before. I can set up these two unique examples here to show you something. Well, first of all, an example of a non-faithful group. So you don't always have to be a faithful group action. And on this side is actually, when we set up Cayley's table, a composition of two elements in a group by that binary operation that all we're talking about really is just group actions. That is really just a group action of the group onto itself and that group being very specifically the group that makes up the set. I mean a set, it just makes up the set that is inside of that group. So that's group actions.