 An important type of permutation group involves multivariable functions. Since we have several input variables, we can permute them. If f of x, y, z, and so on is f of the permutation of x, y, z, and so on, we say that f is invariant under the permutation sigma. The permutations that leave f invariant form a group under composition. For example, let's find the group of permutations on the variables under which f of x, y, z equals x squared plus y squared plus z squared is invariant. Now the function is the sum of the squares of the input variables. Since addition can be performed in any order, then f is invariant under all permutations in S3. What about a function like f of x, y, z, w equals x, y plus z, w? So let's begin by considering what happens if we map x to a later variable. Since our function is a product of x and y with a product of z and w, we'll need to form expressions that include x, y, or y, x, and z, w, or w, z. So if x becomes y, then y must become x in order to get an x, y term. Meanwhile, the variables z and w, we could leave them alone, and that gives us this permutation, or we can switch z and w, and that gives us this permutation. Or if x becomes z, then y must become w, because we need to get that product z, w. And again, z could become x and w could become y, or z could become y and w could become x, or maybe x could become w. Again, in order to get this z, w product, that means y has to become z, and our last two variables can either become x and y, or they could become y and x. Now let's consider anything that permutes y. If y becomes any later variable, x will also have to change, and we've already found all those possibilities, so we don't have to worry about permutations that change y and a later variable. If z becomes w, then w must become z, and we'll leave x and y alone, because we've already considered all permutations that move x. And let's not forget we also have the identity permutation e, and so altogether there are these permutations that leave f invariant. We define a symmetric function to be one whose value is invariant under a permutation of its input variables. Now if a function isn't symmetric, that means some permutations will yield different values. Let's take a look at that. So let's determine if our function is symmetric, and if it's non-symmetric, how many distinct values will it have? Well, we've already determined that this function is invariant under all permutations, and so there's only one function value, namely x squared plus y squared plus z squared. On the other hand, consider something like x, y plus z, w. We already found the group of permutations under which f is invariant, and it's not all permutations, so there are distinct function values. So the distinct values will be the function as defined, because there is a group of permutations that produce the same value. Now to produce a new function value, x has to be multiplied by either z or w, and this leaves only two other possibilities. f of x, z, y, w, that's x, z plus y, w, and f of x, w, y, z, that's x, w plus y, z. And so let's think about this. There is a group of permutations that give us this value, x, y plus z, w, and then every other permutation is going to give us one of these other two values. And this turns out to be very important, and we'll take a look at that in a little while.