 Now, given a g-set s and some element of s, we can form the stabilizer subgroup, gs, and this gives us a new set. Anytime we have a subgroup, we can form the cosets, g, mod, gs. Now, this is not always useful, since the stabilizer might not be a normal subgroup, the quotient group might not be a group. However, it's still a set, and g acts on it in a natural way. If I take an element of g, then g acts on a coset of the stabilizer by ga acting on the stabilizer. And that means this quotient, well, I'll call it a quotient set, is actually another g-set. And at this point, it's useful to develop some intuition. So I'm going to tell you how to have intuition. Well, actually, I can't tell you how to have intuition by definition, but here's some intuition that you might want to have. The stabilizer consists of elements of g that leave s in place. Now, the cosets, g, mod, gs, represent elements of g that move s. That's because our distinct cosets are represented by something that's not in the stabilizer. Now, the orbit of s is all the places s could move to. And so here we have things that move s, and here we have places that s could move to. And so we should feel, and here's our intuition, there should be a relationship between the quotient set g mod gs and the orbit of s. And so the question to ask is, is there a useful function that maps the set of cosets to the orbit? And the answer is, yes. And in fact, there's very natural mapping. Let's consider one of our cosets. Then phi applied to our coset equals as, that's the representative of the coset, defines a mapping from the set of cosets into the orbit. Well, it does have to be well defined. So suppose I have two different representatives for the coset. Since the cosets are the same coset, we know that a prime has to be equal to ag for some g in our stabilizer. So when we apply phi to a prime gs, we get a prime s. Well, that's really a gs, which is a applied to gs. But since g is in our stabilizer, gs is just going to be s. And so this is as, which is phi applied to our coset. And that means that phi is well defined. And this leads to an important result. Let s be a gset with some element s having stabilizer and orbit. Then this function we've defined from the cosets into the orbit is a bijection. Now the fact that this is onto should be obvious. We want to prove that it's one to one, so suppose phi of a gs is phi of a prime gs. Now by our function definition, this means a applied to s is the same as a prime applied to s. And note that these are elements of s. So we can act on them with another group element, say a inverse. And if we do that, we get. And the associativity of the group action means, and now we have operations in our group so we can simplify. And what this means is that since applying a inverse a prime to s just gives us s, then a inverse a prime is an element of the stabilizer of s. Now consider the coset a gs. Since a inverse a prime is an element of our stabilizer, then a times a inverse a prime is an element of our coset. But a times a inverse a prime is a prime. So a prime is an element of our coset. And since a prime is in our coset, then our two cosets must be the same coset. And our function phi is a one to one function. And this leads to an important result. Because there's a bijection from the set of cosets into the orbit, this means their cardinalities are the same. But Lagrange's theorem says that the number of cosets is the order of the group divided by the order of the subgroup. And consequently, if s is a gset with an element having stabilizer gs and orbit os, then the order of the group is the product of the order of the stabilizer and the order of the orbit.