 Welcome back to our lecture series Math 4220 abstract algebra one for students at Southern Utah University As usual, I'll be your professor today. Dr. Andrew Misseldine and lecture 36. We're gonna talk about the idea of Symmetry, all right. What is a symmetry after all? Okay, now get yourself ready for some reading. Yes, Batman said that Batman is very wise He also has huge pecs, but I digress here So recall that the notion of a symmetry is first of all a permutation. So our symmetry here, we have some map sigma which goes from a set X to a set X and So it's a permutation, but why do we have a different word other if we already have permutation the idea is a Symmetry is gonna be a permutation that preserves the underlying structure here of the object X and which case I'm gonna leave this deliberately ambiguous so that we can have a very diverse interpretation for mean by symmetry and to try to illustrate what we mean by this ambiguous definition underlying structure I want to present an example, right? So imagine X is the set. It's first of all, just it's just a set itself So I don't attach to any necessary structure other than it's a set that is its collection of objects Well, as you studied previously in this lecture series if you have some set X then the so-called symmetry group is called S sub X. This is the collection of all permutations from X back into itself If X has no other structure, then the symmetry group S sub X captures all of the so-called symmetries of This of the set X. So for example, if X is just the set of numbers one two three four Then we would call it symmetry group S4 and when I say that it's just a set I mean I have no no interpretation of these things other than it just I have these four objects, right? One two three four and so by this symmetry group I can take any permutation where I scramble up the one two three four So you could have like one goes to two two goes to three three goes to four and four goes to one That would be an acceptable symmetry of this set We could have that one goes to two two goes back to one three goes to four four goes to three That would be acceptable you could have that one goes to two two goes to one three goes back to itself and four gets back to itself That's a symmetry we could have as the symmetry just the identity one goes to one two goes to two Three goes to three and four goes to four. These are all acceptable symmetries on this set Which again, it's just the numbers one two three four. We put no structure on this set whatsoever But maybe x does have some structure some other structure than just being a set, right? There's something on top of it So for example, let's take the exact same set as before x equals one two three four But these four points could in fact just be labels of the four vertices of a square as you see illustrated over here So that is we just have the four corners of a square then We're not going to consider every possible permutation of x anymore Because not all of those permutations would be permutations of the square what I mean by square, right? What I mean is there's adjacency with associated to the square, right? The number one is adjacent to two and three as if we view them as vertices of the square And likewise the number four if we think that it was the fourth label of this square Then it's adjacent to two and three, but it's not adjacent to one, right? There's something different between four's relationship to number two and number three Compared to its relationship with number one That is you don't have this adjacency relation Attached to one and four likewise you don't have an adjacency relationship to two and three In some regard they're on opposite sides of the square and the geometric Or you could call it the combinatorial structure of this square Then suggest there's something different between one and four versus its adjacent pairs two and three So we wouldn't consider all of the permutations on the set x But only those permutations would preserve this adjacency relation That is only those permutations have preserved the shape of the square And of course we've studied this in detail previously in this lecture series The only permutations Of the the only permutations that would be preserving the squareness Of x is going to be d4 Which d4 was defined to be the symmetries of the square and this kind of leads itself to Other situations, right in combinatorics one is often interested in the idea of graphs Where a graph is some collection of vertices So you just have these points which we could label them whatever we want and then Then points or vertices of the graph are connected to each other by so-called edges And then you could talk about this adjacency relationship So we could have something like the following we have vertex one two Say following the same scheme here three and four for which we could say that one is related to four Two is related to To three and that's it if that's what we wanted then in that situation we would only preserve those We'd only take those permutations that preserve the structure of this x right what if we have an x as our shape Which is a different type of graph well in this situation you could take One and four could swap locations Um two and three could be left alone in which case then okay Then in that case you just kind of switch the labels on the The what we want to call that the the slant that goes from top left to bottom right you could do that That would give you a symmetry of the x and that would just be the one-fourth symmetry right there We could also swap locations of two and three we get that two and three we could swap swap locations of two and four and Three two and three together right so you could put them together as the two-two cycle I want you to be aware that That permutation we could talk about the switching switching to spots We could also think about like what if I kind of rotated my square Maybe I could draw a little bit more symmetrical so it looks more like maybe like a cross or a plus or something like this Right, you could do like a you know rotation of some kind like that Then you could think of these as like reflections and rotations Um throw in the identity into that mix as well In which case then we see that the symmetry type of this cross x plus whatever you want to call it It gets the symmetry type of the client four group right And then we could keep on going from this right we could put my pictures back here. We could take We could take oh, let's say that somehow one Two and four related to each other, but three is somewhat isolated Distinguished from everything else and it wouldn't be so ridiculous to do something like that But let's say that we had uh, you know this polygon or not polygon, but this graph that did something like this Then it makes sense to be like, okay, maybe the points one two and four are completely interchangeable You get this symmetry type of s3, but there's something just something distinct about Three right there's something different about all of it and there again there there could be Quite reasonable ways to make sense out of this right you could think of it like oh maybe like one two and four Are points that live in some type of plane, right? But then number three was some third point that lives outside of the plane And so it's distinguished from all the others for some reason. I don't know And so depends on the structure of the set and so i'm giving you some examples of geometric slash combinatorial Relations structure we could place on the set, but we could also be asking ourselves What type of algebraic structure could we place on the set one two three four, right? We could ask ourselves If I take if I take the set one two three four to be the sicklet group of order four, right? So these these numbers are elements of a group one two three four Where we'll say that one is the identity one two three four We get and then the kately table is just displayed right here We'll say that two times two is three Two times three is four two times four is one we'll get three times two is four three times Three is one three times four is two and like lies four times two is one Four times three is two and four times four is three So in particular if you want to think of this in the added additive sense Then multiplying by two with respect to this kately table is like adding one mod four It's a two would be the generator of this group So we have this sicklet group of order four It's kately table is given so it is it's like I said it's isomorphic to z four In this situation the symmetries of x if we view it as a group These need to be permutations that preserve the group structure Uh, that is that because that's the underlying group in play right here Well, if it preserves group structure, then we need to be it needs to be a homomorphism phi of ab Needs to be equal to phi of a times phi of b So we need it's a it's a permutation, right? We want permutation. So that means it's going to be bijective It's going to be bijective, but we also need it to be homomorphic so it preserves the group structure So we have a a bijective homomorphism. That's what we normally call an isomorphism But as this is an isomorphism from the group back into itself in the literature This is what's referred to as an auto morphism So morphism like it usually means in the set in this setting here It's like it's something that measures the shape morph means shape here auto actually means self Right. So like an automobile is a self moving object, which in the united states We usually just call that a car, right? But if we're looking for the symmetries of a group, then these are called Automorphisms of the group. So let me kind of emphasize that for a group structure The symmetries are these automorphisms or as I was above talking about those graphs and geometric shapes before we might talk about the automorphisms of the graph Well, what automorphisms are there for the group? So what permutations can you can you what permutations allow you to move things around? But you still have the same underlying group structure that you started with Well, in some regard when you have the identity right here, right? We know from previous study that any homomorphism which would include iso and automorphisms as well Any homomorphism will send the identity itself kind of like I was mentioned earlier, right with that graphical example The number one because it's the identity is is somewhat distinguishable from every other element, right? You can tell the identity apart if I just kind of grabbed an element from random Randomly from the group like I just imagine my group is like this bag of marbles And I just select one marble from the bag You can tell if you grab the identity marble because how it acts with all the other elements of the group also this element three I would also argue is distinguishable from every other element of the group because with the exception of the identity The element three is the only element of this group which has order two That is when you take the element you square it three times three gives you one in this situation You can't tell that apart between two and two and four, right? Now because remember you grab an element so it's like if I grab a marble from my bag There's these four marbles labeled one two three four, but I don't see the label I just have to kind of experiment with other elements. Well, okay if I grab one element one marble I can multiply it by the other marbles and see oh it's acting like an identity therefore it's going to be the identity I can tell who the identity is With the element three right here if I grabbed it's like okay. I could tell that three is not the identity But I can also tell that three has order one. Oh, this has to be the number three. I can figure that out Because again, there's no label placed on and we just have I guess what I'm trying to say is like What if we encoded like these marbles, right? What if I what if I gave you a marble of like blue Red green yellow something like that and you have to figure out what it is how it acts in the kelly table When you can figure out that the blue marble gives you when you multiply by another marble gives you the other ones You can find out that the green marble always gives you one But when you get like the red and yellow marble when you look at two and four right here The thing is you can't tell them apart from the algebraic sense that I could interchange I could interchange the numbers two and four on the kelly table and there would be no loss whatsoever Let's try it out real quick. If I just write every two I'm going to put a four Every two becomes a four like so and then every four is going to become a two You get something like that. We could try this again, right? What happens? Well, one times one is one one times four is four one times three is 31 times two is two. That's great Um, if I jump to the bottom here, you're going to get two times one is one two times four is one Excuse me. Two times one is two two times four is one two times three is four two times two is three Okay, um, and then you go through all the rest of them. It's the exact same kelly table now Yes, I've scrambled up the rows and columns, but that's what permutations do don't they they can scramble things up But even though it's scrambled around it's still the same It's the same kelly table. And so this automorphism is in fact a It's a symmetry of the group and so the so-called the so-called Automorphism group that is the symmetry group of this of this group So the automorphism group is a is a symmetry group of groups You're going to get two symmetries. You have the identity which always works But you also have this relabel and why does it say one three that should say sorry that should say two four One and three are distinguishable there two and four that permutations we can move around But we could also consider like the climb four group for example That's a group of order four and so maybe we take it as like one two three four One two three four Right, so the identity will be will be one And then everyone else is going to be order two so when you square it you give one So you're going to have and the way the climb four group Has when you have the elements which are non-identity when you multiply them together you always get the other element So two and three gives you four two and four gives you three Three and two gives you four three and four gives you two four and two gives you three And four and three gives you two like so And so with this kelly table if I kind of lost the labels You'd still it'll tell the identity is the identity because when you act on it by any other element you would get That element back But the thing is when it comes to two three and four if I relabeled them If I relabeled the two threes and fours you get the exact same group back again And so while the automorphism group of say z four This is just isomorphic to z two you only have the identity in that that two cycle when it comes to the automorphism groups of the climb four group here Right, and this actually gives you s three because you could take any permutation of Of any permutation of the elements two three four and that would give you back the same group again So when we view x as a set We got the symmetry just to be s four Okay, because there was no structure to it when we viewed it as a square We got the dihedral group d four which is also a subgroup of s four mind you right And we could take a different shape or different graph attached to just attach sort of a geometric meaning to it We get these other symmetry groups which are based upon the geometry of the set When we viewed x as a cyclic group then the symmetry the symmetries turned out to be isomorphic to z two Which would be the automorphism group. This is also right z two is a subgroup of s four If we thought of as the climb four group you get s four Which s four can also be that excuse me s three which can also naturally be viewed as a subgroup of s four So in this manner When we consider the symmetry group of an object we need to focus not just on the elements Which create the object, but also the relations between The elements that capture the aspects of what we're focusing on Is the set combinatorial is the set topological is the set geometric Is the set algebraic is the set number theoretic What have you and so these sets often have structure to it? Is this an ordered set? Is it a partially ordered set just to name a few examples, right? Is this a partition this set has structure? And so as we talk about symmetries the symmetry group of a set it depends on the underlying structure geometric algebraic What have you and so when you focus on those aspects you'll be able to determine what type of symmetries Am I concerned about but in all aspects the symmetries will always turn out to form a group You