 So let's see if we can study a group that is somewhat more concrete in some sense, and that's the permutation group. So let's take a look at this. So given some set of distinct symbols, a permutation consists of some rearrangement of those symbols. So for example, if I take the symbols a, b, c, and d, I can rearrange these symbols as a, c, d, and b. Now one of the questions that mathematicians always ask is, hey, I have this great idea, how am I going to write it down? And so one of the ways we have of expressing a permutation is to use what's called Cauchy notation. And that's named after this guy, Agostin Louis Cauchy, who lived during the 18th century. This image is from the Mac Puder history of mathematics archive, very, very useful site if you want to know anything about the history of mathematics. Now Cauchy notation looks something like this. I can take my set of symbols a, b, c, and d, and here's how I've rearranged them as a, c, d, and b. And the proper way we should actually read this is I've taken a and replaced it with a. I haven't done anything in other words. I've taken b, replaced it with c, c has been replaced with d, d has been replaced with b. And one important thing to think about here is that it's the permutation is not as the rearrangement, but it's not actually the arrangement of the symbols. This particular arrangement, a, c, d, and b, is only here because this arrangement was a, b, c, and d. What I've really done is I've left a alone, I've replaced b with c, c with d, d with b. If I were to go through this entire text and do that replacement, b with c, c with d, d with b, that would actually be a permutation. This idea is important to keep in mind as we move through our discussion of permutation groups. Again, you want to think about permutations as replacements of one object with another. One final note here is that a permutation does not require replacing every element with something else. If you go into study combinatorics, you do study permutations that do require this replacement, and these are called derangements, which is the term for what happens after you study these for two lines. There's a couple more terms that are useful to describe, which are the following. If I have a permutation, so sigma and tau, for example, the support of sigma is the set of elements that are actually moved around by that permutation. To permutation sigma and tau, we say they're disjoint if their supports are disjoint. For example, if I have these permutations, sigma in Cauchy notation, tau in Cauchy notation, and sigma has supports b and d, because these are the things that are moved around. a, c, and e all stay in the same place. Likewise, tau has support c and e, because these are moved and a, b, and d don't change. And since their supports are disjoint, b and d, e and c, these permutations are disjoint permutations. So one of the important goals of abstract algebra is to thinking like a mathematician. And so what we have in front of us is we have these set of permutations of n objects. Now if I take this entire set, what I'm going to do is I'm going to designate that entire set Pn. So Pn is this set of permutations of n objects. And given a set, one of the first things a mathematician might ask is, how can I turn this into a group? So in order to become a group, we need a couple of things. First of all, we need a binary operation. I need some way of taking two permutations in this group and combining them, doing something with them, mixing them with i of nu to whatever, and producing another permutation. And then once I have that binary operation, I need to have the other group properties. So what can we do? Thinking, thinking, thinking, thinking. The binary operation we'll use for permutations is called the composition. So if I have two permutations, sigma and tau, the composition sigma and tau, we're going to define that as the permutation that we get by performing tau first, then sigma. And one important thing to note here is that we have this reversal of order. I read this as sigma tau, but the way that I should think about it is tau is the permutation that's applied first, and sigma is applied to whatever I end up with. And this is an inheritance from thinking of the composition as applying to some arrangement A. And so if I were to use function notation, I'd have A, then Q, then P, and which indicates that I apply Q to A first, then I apply P to whatever the thing is. And so that's why we have that reversal of order. All right, so let's take a look at one composition. So here I have two permutations. Sigma looks like this, tau looks like this, and let's see what the composition sigma tau and tau sigma look like. So to compose permutations, I'm going to follow the path. So if I want to find sigma tau, I apply tau first, then I apply sigma. So here's my initial set of objects, A, B, C, D, and E. And if I apply tau first, A gets replaced with B. And then if I apply sigma, B gets replaced with E. So here's why it's important to think about the permutations as a replacement and not a rearrangement. B is going to be replaced with E, and so that's where I end up. And B, I'll apply tau first, B gets replaced with C. I'll apply sigma, C gets replaced with B. C, applying tau first, C gets replaced with A. Applying sigma A gets replaced with A. It doesn't change. D, apply tau first, gets replaced with D. No change. D gets replaced with C. E, no change. E gets replaced with D. And so there's our permutation, tau, then sigma. This A, B, C, D, E gets replaced with E, B, A, C, D. Now we don't actually need that middle row that was only necessary for us to figure out where we end up with. So we can actually, and should, if we want to write this in Cauchy notation, we'll drop out the middle. And so our permutation, sigma tau looks like this. Likewise, tau sigma can be found in the same way. So here's tau sigma. I'm going to apply sigma first, then I'm going to apply tau. So A, applied to sigma A, gets replaced with A. And then tau A gets replaced with B. B gets replaced with E. And then E stays the same. C applies sigma first, gets replaced with B. Apply tau now. B gets replaced with C. D gets replaced with C. C gets replaced with A. E gets replaced with D. D stays the same, and dropping out the middle row gives us the product tau sigma. And one of the most important things to notice here is these are not the same permutation. And that means that permutations do not commute. Here we have a nice set of objects with a binary operation, and this binary operation is not commutative. Permutations are very abstract and can be very complicated, and you might wonder, well, why are we studying them? They seem a little bit useless. And so the reason that permutation groups are very important is the following. Suppose I take a group with four elements, for example, I have some identity A, and I have some other elements B, C, and D, and these are all assumed distinct. So we could represent this group using a Cayley multiplication table, and so maybe it looks something like this. And if you look closely at this table, one of the things that you might notice here is that each row of the table is a permutation of the four elements A, B, C, and D. So what that means is that this group, this Cayley multiplication table, looks an awful lot like the set of permutations, these. And this is a subset of PN, and if I use the operation of composition, this subset actually forms a group. And what that means is that every group of four elements, every group of N elements, regardless of what that looks like, regardless of what its Cayley multiplication table is, every group of N elements corresponds to a subgroup of PN with N elements. And what that tells me is that if I know everything there is to know about permutation groups, I actually know everything there is to know about every group that exists. And so permutation groups are extremely, supremely important.