 Given a group G and a set S, a group action F from the Cartesian product G by S into S defines two important objects. Given an element of S, we can talk about the orbit of S, which we talked about previously, and also the stabilizer. Let F be our group action for an element of S. The stabilizer is the subset of G, where G applied to S just returns S. If the orbit answers the question, where in our set S can I go, the stabilizer answers the question, what in our group G will keep me in place? So for example, let's consider the multiplicative group of integers mod 31 and S to be the set of cosets of the subgroup generated by the element 2. And let's find the stabilizer of the element 3H. And that means we want to find elements of our group where G applied to the element 3H gives us the element 3H. So to begin with, we note that 3H is the set. And also because this is a coset, our cosets are either disjoint or identical. And so if an element of A applied to 3H, which is to say this set is already in 3H, then A is going to be an element of the stabilizer. So let's consider, maybe 3A is this first element 3, in which case A is 1. And so 1 is an element of our stabilizer. Or maybe this element 3A is 6, and we solve and find. And so 2 is another element of our stabilizer. And similarly, maybe 3A is 12, giving us 4 as an element of the stabilizer. Maybe 3A is this element 24, and so 8 is an element of the stabilizer. Or maybe 3A is 17, and that gives us 16 as an element of our stabilizer. Now if you think about this carefully, you may wonder, do we also need to solve 6A? Is 3, 6, 12, and so on. And it turns out, we don't have to, we just get the same solutions as we did before. Or again, if our set consists of different expressions involving four variables, we might try to find the stabilizer of a particular expression about x, y, plus z, w. And so we note that we're looking for the permutations that leave our expression essentially the same, where we assume that our variables still have the same properties as the real numbers. And so we might do nothing, that's our identity permutation. If we swap the variables x and y, that doesn't change anything. If we swap the variables z and w, that also changes nothing. And if we swap both sets of variables, that changes nothing. And so our stabilizer is going to be the set consisting of the elements, the identity, x, y, z, w, and x, y, z, w. Now by definition, the identity of g acting on an element of s gives us s, so the identity of g is an element of the stabilizer. And since the elements of the stabilizer are elements of g, they're subject to the binary operation of g. And so the question you've got to ask yourself is, self, might they form a group? And in fact, they do let f be a group action. For any element of s, the stabilizer of s is a subgroup of g.