 Given a group G and a set S, we can define a group action. This is a function from G cross S into S with following properties. The identity property, if E is the identity in our group G, then ES is equal to S for all S in S. And associativity, if AB are elements of G, then AB applied to S is A applied to B applied to S. If a set S has a group action via this group G, we call S a G set. Now, we've actually seen these group actions before, although we haven't called them that. For example, if H is a normal subgroup of G and S is a set of cosets of H, then F from the Cartesian product into S, where F of G of AH is GAH, is a group action. Well, don't believe me, we'll verify those requirements. So first, we need to see what happens if we apply F to the identity and a coset. So first of all, by the definition of our function F, F applied to EAH is E applied to AH. Well, we note that E applied to AH less just a coset formed when the elements of AH are multiplied by E. But remember, these are the identity and group elements, and so when we multiply an element by E, we just get the elements of AH. Next, we want to verify associativity. So we need to check F of PQ applied to AH. So again, we note that PQ applied to AH is a coset formed when the elements of AH are multiplied by the product PQ. So these are elements of the form PQAH, where H is an element of H. Now, since the group operation is associative, we know that PQ AH is the same as PQAH, and so these products will be elements of P applied to QAH, and so associativity is also verified. Now, once we define these group actions, we can consider the following ideas beginning with an orbit. So given some element S in our set, we define the orbit as follows. It's the elements of our set S, where X is equal to GS for some G in our group G. Now, it's important to note that the orbit lives in our set S, but it's based on elements in our group G. So for example, let's consider the multiplicative group of integers mod 31 and the set of cosets of our cyclic group generated by the element 2. And let's find the orbit of the element 3H. So first, we find this subgroup that's generated by the element 2, and this is going to be. And now we can find our cosets in the usual fashion, and so we find. And this collection of cosets is going to form what we call S. Now, notice that the elements of S are subsets of the elements of our group, and this is an idea that will become important later. So remember the cosets are either identical or disjoint, and so we can try to produce a coset AH by multiplying 3 by something to get A. And so we note that 3H itself is going to be. And so obviously we can get to 3H, so 3H is in our orbit. To get H from 3H, we want to find X so that when I multiply 3 by X, I get 1. But since we're living in a group, we can solve this and we find X is 21, and so 21 times 3H is going to be H. And since H is something we can get to if we start at 3H, then H is going to be in the orbit. Similarly, if I want to get 5H from 3H, I want to find X so that 3 multiplied by X is going to give you 5. And again, because these are group elements, we have a solution X equal to 12, and so 12 times 3H is going to give us 5H. And since we can get to 5H from 3H, then 5H will also be in the orbit. And continuing in this fashion, we see that we can generate 7H, 11H, and 15H by an appropriate choice of multipliers of 3H. And so we see that the orbit of 3H is going to include all of our cosets. It's going to be our set S. Historically, these group actions are among the oldest parts of group theory. They appeared in Lagrange's work on the resolution of equations, and he applied them to the cases of multivariable functions and a permutation of the inputs. So let's consider our expressions that we can be formed by permutations of X, Y, Z, and W in X, Y, plus Z, W. So there's our set, and let's consider a group of permutations on 3 elements, X, Y, and Z. And so let's find the orbit of this element, X, Y, plus Z, W. And here we're assuming that our variables have all the properties of the real numbers. So we'll abuse notation a little bit and write sigma applied to X, Y, plus Z, W to indicate that we're applying the permutation sigma to X, Y, plus Z, W. Now, since we're dealing with sigma from the group of permutations on 3 elements, then sigma will only permute the elements X, Y, and Z. And then the elements of S we can obtain by applying sigma are the permutations of X, Y, and Z in X, Y, plus Z, W. And these are. And note that any other permutation on X, Y, and Z will produce one of these expressions. And so we find the orbit of X, Y, plus Z, W to consist of the 3 distinct elements. Again, it's important to remember that the orbit of an element of S is a subset of S. And so we note the following theorem. Let F be a group of action, and suppose S and S are primary elements of S. If the orbits have a non-empty intersection, they must be identical. Consequently, our set S can be partitioned into distinct orbits. But since S is not a set, there's not a lot we can do with this partition, at least not yet. Let's build up a few more ideas and abstract algebra.