I'm sorry I was a little late in putting the homeworks up. I did put them up. Let me know if it's a problem. It is a problem. Yeah. I even put the M-sweeps up at the start, but I made changes. So anyway, they're there now. You can do them. We can postpone the deadline. We can postpone the deadline if you want, but I can't take the other one. Let me know if it's a problem. We have some function matrix of such a function, which is the matrix of partial derivatives. We have the gradient vectors of each of the coordinate functions, or going down we get tangent vectors like this. So as an example, there could be only two examples. Suppose I have a function from r3, r3, and since I used this twice in my examples, x squared plus e to the y, of course I'll forget it by the time I get there. f of x, did I have sine or cosine? I think it doesn't matter. Sine of z plus y. And so we can compute. So the book uses the notation here f prime. I really like using the f prime notation because people tend to think that this is a one-dimensional thing. They tend to use capital D to make it clearer that it's a derivative matrix. Of course, in all of these cases, just about everything in sight is a vector, not everything. So anyway, I'm going to continue to use the df notation rather than f prime. But in this case, then our derivative matrix here will be going across. I'll get 2x. Can I have an x there? No, I guess not. OK. Yes, I did. Huh. No, I guess I did. OK. 2x e to the y, zero. The derivative of this guy with respect to x, y, and z. The derivative of this guy with respect to x, y, and z gives me 1. With respect to y, I get sine of z. With respect to z, I get y cosine of z. And then here I get 1, 1, and 0. And so at the point, so here we see that f of 1, 1, pi is 1, 1, pi is 1 plus e. And so here, put this function at 1, 1, pi, 2, 0, right? Cosine of pi is minus 1, 1, 1, 0. Right? So this is the linear map that approximates f or near the point 1, 1, pi. And so then you can check if you want that the gradient of this at this point is 2d0. The gradient of this at this point is 1, 0, minus 1. The gradient of this at this point is 1, 1, 0. So everything's good. And, okay, so that's just to remind you how all this stuff works. Any questions about that stuff? Everybody's good with that, right? I'm just sort of resetting the stage. Okay. And then where I left off, right at the end of the class I talked about the tangent approximation. So f there. So if I have this same function f of the tangent approximation to f, which is sort of the first term of the Taylor series, it's just going to be evaluate your function at some point. And then we add on the derivative matrix evaluated at that point, times where I think of this as a column. So this is just saying to me close to approximation or in one variable calculus. This is the tangent line approximation, except it's not a line, right? So I think the same example, if I wanted to estimate, no, but I guess I'll leave this lower. So if I want to estimate, say, f of 1.1. Can you say tangent when we have that next one? I see. It's not really two-dimensional. Tangent. Let me just say tangent. Yes. It's a hyperpoint, right? Right. I mean it's, yeah, it's a hyperpoint. Right. Here I have three, so this has to be, it's the tangent space. I just lost my place. Okay. So say I want to approximate this. This is going to be just about f of 1.1 pi, which is right there, plus this makes minus 1, 1, 1, 0, times the column vector, which is the difference between this and this. This will be 0.1 in the x-coordinate, minus 0.1 in the y-coordinate, and 0.01 in the z-coordinate, which is, we lost it.  That's the change in x. The change in y will be 0.1 minus 0.01, and the change in z will be 0. It will be 0.1 minus 0.1 plus 0. So that means this will be 1.2 plus e over 10. No. Sorry. That's 0.9e. That's from this. And here I get, that's 0.09, that's 0.2.  And if I am approximating e on a calculator, then I can certainly just compute the value which I did, 3.669, 1.09. So again, this is just the same stuff that you did in one variable calculus where you want to approximate the cosine of 0.3 or 0.1 by looking at the derivative, looking at the value of the cosine at 0 plus 0.1 times the derivative. That's exactly the same except here we're moving using the tangent approximation. Let me do one more example. And this example is in the book except not really because they say it's something that it isn't. So if we take, say, g of uv, so this is going to take n, I'm going to take 1 something away from 0. Well, let me write it down. So which do I use? Cosine or sine?  So the book claims, the book does the example with a u here and then claims it's what I'm going to draw but it's actually not. It's a comb if you put a u here. But it says that it's a ramp which is just wrong. But it does everything else right so it doesn't matter. So something that we'll take, say, in the uv plane and I map it into yz space, then for a fixed value of u I get a line pointing at the origin and for fixed values of u I get something that spirals up. I get helixes. For fixed values of u, if I'm going to fix u, then I'll get helical things going up. Then for fixed values of v I get lines. So what I get is a ramp-like structure. Can you even see what I drew? So I get a spiraling up thing like that. Now it has a problem at 0 because at 0 it stretches way out. But other than that we're okay. So I get this ramp-like structure. Now let's look at what df-dg tells us. Let's just compute dg. So dg will just be the, well I have two variables in and three variables out. So I'm going to get a 2 by 3 matrix or 3 by 2. Let's forget who's a row and who's a column. But anyway I get the 3 by 2 matrix. So if I take the partial of u cosine v with respect to u, I get the cosine of v. If I take the partial with respect to v, I get u in a little more space. Negative u sine v. If I take the partial of u sine v with respect to u, I get the sine of v. No, it's okay. And with respect to u, I get u cosine v dot cove over v. And if I take this, I get a 0 and a 1. So I get that. 3 by 2 matrix. And let's say at the point which is the image which will be sort of here somewhere. Make a quarter term. Then let's see what we get. So dg of 2 pi over 2. We'll be in the 3 by 2 matrix. So the cosine of pi over 2 is 0. The sine of pi over 2 is 1. So I get minus 2. The sine of pi over 2 is 1. So I get 1. The cosine of pi over 2 is 0 again. And I get a 0, 1 there. So I should get that matrix. And now what is that telling me? Well, certainly I can do the same business here where I can think about this as an approximation. But I maybe want to interpret this slightly differently. If I multiply this... Well, no, let's... So now let's think about how I can see the tangent plane from this. So here, if I'm looking at the image of the point 2 pi over 2. So g of 2 pi over 2 is the point... x is... Uh-oh, where's my map? x is 0. y is 2. And z is 2. So near 0, 2, 2. I want to claim, so near 0, 2, 2. My tangent plane... Did I make a mistake? Yeah. Is it moderate to the last value? Yeah. Yeah? Yeah. Oh, yeah, yeah, yeah, sorry. My tangent plane, what does it look like? Let's think of it as being attached here at the point 0, 2, pi over 2. So it looks like... I mean, I'm at the point 0, 2, pi over 2. And then these columns tell me the parametric representation of the tangent plane. So it's going to be 0, 1, 0 times some vector. The x direction... t. Well, if I move in the u, if I move u, then I increase... Well, let me just call it t for a moment. And I give it 2, 0, 1. Right? So this is saying go to the point 0, 2, 0, 2. In one direction, increase... And this is increase v. Well, we shouldn't have used v. It's just the same statement, just interpreted in a different way. This is telling me that near the point 0, 2, 0, I have a tangent plane which contains one vector that only points in the y direction, and that is the image of the u-axis. If I increase u and hold y fixed, then I move this way in the y direction. So I imagine there's z. That's z. Here's y. Here's x. So if I increase here in this direction and move in this line, which is parallel to the y, and exactly at the rate in which I increased it, and if I increase in the other direction, here... Which moves... Initial graph, 0, 2, higher than 0? No, my initial graph goes to, like, 3 pi. So you're... What do you mark as an initial graph? Yeah. It doesn't look like 0, 2, 0. Because 0 and x is not... Oh, sorry. Okay? Yeah, okay. Why did I want that to be possible? Oh, no, no, that's right. So if I move a little bit in the u direction, I mean, fixing u and moving in the v direction, what happens to the x? Well, it decreases a lot, but the z increases. So this is a vector pointing up. So I can see this tangent plane sitting there. I don't know if you can see it, but I can see it. It's tilted up, so that the x decreases by a factor of minus 2, and as the x decreases by a factor of minus 2, the y doesn't change, and the z increases by a factor of s, however much I move the other guy. And, you know, if I did this at another point, which was turned a little further, I would see you get some combination of these things turning a little more. Is this clear to everybody? Am I meeting dead horses? I'm not sure. Okay, so I was talking to another professor who was telling me that, you know, she can't always read people's faces.  And right now, I thought, yeah, I don't ever see that, but right now, that's what I'm seeing. It's like... So I'm going to assume, unless somebody tells me that it's the, yeah, whatever, get on with it. Yeah. So that's the tangent plane. Are you going to point the gradient? Does it follow the spiral up? No. That's what I want to talk about next. No, it's perfect. Great leading. What about the gradient? Well, here, see, I don't really have a gradient. I have several gradients. Because this is not a function. This is a function from R2 to R3. So there's two gradients. Because there are, there's three gradients. I, I have multiple functions involved. The gradient only works for a function into R. Right, a function where I put in a bunch of numbers and out comes one number. Here I'm putting in two numbers and out comes three numbers. The gradient is here. There's actually three gradients. There's the gradient of that function, the gradient of that function and the gradient of that function. And those correspond here, here and here. There are three gradient vectors laying around here. There is the vector zero minus two. There is the vector one zero and there is the vector zero one. Exactly what those mean, I want to put off for just a few minutes. But there are three gradient vectors here. But then there are sort of three tangent, two tangent vectors which give me the tangent plane that I can interpret in this way. So depending on how I want to look at this matrix, whether I want to pay attention to the columns or the rows, tell me whether I'm looking at gradients or whether I'm looking at tangent vectors. It's a little harder in the case of this three by three matrix. The tangent vectors are there but they're three dimensional so it's a little harder to see something tangent to a surface because they don't really have a surface. It's a space that is being used there. Okay? So I'll say more about gradient vectors and stuff real soon. Are we okay with all this? So now I'm going to start new stuff. So this was review, although really I kind of stopped about here last time. So it's review of what I didn't talk about as opposed to review of what it is. Whoa, whoa. Okay. So for a few minutes, no, maybe not yet. So now I want to specialize once again to a specific class of functions. Suppose that my input and my output spaces are of the same size. So I have a function from R n to R n. I can think of that and I'm going to mostly think of it as n is 2 or maybe n is 3. So one way, and let's see, I have an example function that I was dragging around 2x, y over 2. And let me just use a calculator for that. So f of x, y is 2x y over 2. So one way we can think of this and this is the way that we've been thinking about this so far is I put in a box. I apply f to it and out comes some shape. In this case, since I'm doubling the x coordinate this box will get twice as long and I'm having the y coordinate so this box will get skin here. This output will be something that looks it'll just take this box it'll squish it in the y direction because the y goes to y over 2 and it'll stretch it in the x direction. I can do things more complicated where I'm going to mix up the x's and the y's and all sorts of stuff but maybe this is enough because it's easy to see what that function does. So this is one way to think of a function of two variables. But another way that is very useful and this is in fact a linear function two variables because I wrote a linear function you can think of a function from r in to r in which somebody asked me when I first did it in Q said well isn't that a vector field so yes now I want to tell you what a vector field is so this is one way this is a mapping which takes in some domain and it does stuff to it and I get something out but another way that I can think of this at least in the case where and the input and the output are the same dimension is I can think of this now here what this means is rather than drawing a picture of the input space and a picture of the output space what I want to do is to each point x y I want to attach the concept works no matter what dimension I'm in whether I'm in one dimension or eight dimensions but I'm going to draw it in two because two is easiest to draw now so that means that really in terms of the transformation I am not connecting I'm not thinking of take this box and draw an input from the source point to the output point instead I'm thinking of this as sort of giving me an infinitesimal motion of each point so here like along the x axis again let's use the same x y is two x y over two so along the x axis there's the zero vector attach that's pretty boring but as I move along the x axis say at the point one I attach a vector from point two and it points at the point negative one I attach a vector of going negative two it points outward at one half well it's too long I attach a vector of size one and so on so as I move out here and they point away from zero and similarly along the y axis I attach vectors which get bigger and bigger but they don't get as big as fast in the x so they're a little shorter and then because of the nice fact that this is a linear a linear function these vectors are very simple and they're just a combination here so out here at a point like let's do one one and a point one one the vector that I attach is the vector two one half so that'll be something like that and in general I will get a bunch of vectors as I move further in the x direction and moving up they sort of fall over and point more in the x okay now really this vector field is describing the derivative of some function but so this is a different object than this but I want to eventually try to relate these things these vector fields are very important in differential equations and modeling things but we'll come back to that when we get to it we're not operating on vector stuff but a lot of this course we'll come back to vector fields a lot because they are an important part of what we're dealing with and I guess so let me just say this now you can imagine I can find a curve this is not what we're going to focus on we'll focus on this in a little while if I have some initial point I can find a curve where the vectors are tangent this is something that one often does and you want to do where the vector field describes something about a function from describe something about a curve this is solving a differential equation this is also called integrating the vector field where we want to find a curve where the vector field that is described here looks nice to them so I'm not going to talk about general vector fields now but the concept of general vector field the sort of thing many people have seen vector fields before who hasn't seen a vector field before ok so we'll talk about that in some cases and in particular I want to talk about special vector fields which is a gradient vector field so here I'm going to start with some function f and then if I take the gradient of f if I look at so in this example f we know this defines for us the surface right so in this case and elongated in the x direction and shorter in the y direction so we'll get sort of a parabolic thing level curve I would get those are supposed to be ellipses and they're supposed to get closer together confocal ellipses it doesn't look very good I want to consider now instead of this function if I look at the gradient of f focus on this example for a minute the gradient is a function from r2 r is a function r2 to r2 this f that I have here is a derivative thing is we okay with this? there should be some very clear relationship between which it is is it clear what it is? I mean if I put of course these are on different scales but if I put this on top of this notice that arrows along the x axis well if I could draw arrows along the x axis that way yes exactly so the thing that maybe this picture should have in this example and it's true in general the level curves taking r on some open set u if it's the zero vector I can't I mean it's still orthogonal but I want to interpret something here so for each guy in there increase make it a curly u you can call it d like the book does I don't like the color it's just a blob that is the domain if you want you can just say differentiable near x each appropriate you can take the derivative around x then you have a tangent plane there and this gradient points where the tangent plane goes up the most and furthermore is the slope in the direction it's the amount were you asking that I'd say do you measure curves now sure well you can prove that it always points in the opposite direction of the normal line and the normal line points in the direction of the lowest so the greatest direction how do I prove that it points in the opposite direction of the normal line because if you think about the function of the tangent it's f of x where it's f of x times x plus f of y times y plus d times something like that so the normal line to the tangent line at that point is going to be f of x f of y derivative of f taking the x direction derivative of f taking in the y direction 1 and then the dimensions you're in and then the gradient looks really takes the x and y part of that and is in the, or for a two-dimensional case would be f of x and f of y so it's pointing in the opposite direction because the normal line is f of x or f of x, f of y and the negative one and then the opposite direction of that is just going to be f of x f of the x, f of the y direction and since that's always pointing in the direction of the lowest increase that means that the gradient vector will always be pointing in the direction of the greatest increase so let's do it a different way it's exactly the same way it's actually what you said but so I'm going to do it in some way that looks different but it's in fact exactly the same so I'm going to let you be a unit vector people understood what he said was not wrong he was right he was saying you look at the tangent plane you write the equation for the tangent plane and you forget about the z component of the tangent plane and you see that the tangent plane corresponds to the gradient and then stuck up in the z component and then you just project that and you see it goes the other way so everything's cool that's essentially what he just said and this is almost exactly the same except it isn't so I take any unit vector and let's compute the directional let's compute I'm going to take the directional derivative unit vector so this is at some point so just pick any old direction that you want to go in and take a unit vector that points in this and we saw before that that is the gradient evaluated at x dotted into the formula that we came up with you want to take the derivative in the u direction that's what it is so now this will have its maximum gradient vector here some unit vector u here and I'm taking the dot product which gives me the piece in the u direction this will be the biggest one in the parallel so that means that u is pointing now in the direction of maximal increase so let's say that sign of the angle between them but if we are and so the factor here the increase factor the length also then this is the maximal since here theta is negative there's nothing going on here but in some sense there's a lot going on the proof is like take the derivative dot in particular that tells us that these arrows are orthogonal I'll come back to that but they point in the way that the steepest way up if I look at this surface here and I take some point then the gradient is going to point the steepest way up the points help the most yeah then the function didn't happen to be differential I don't know maybe like two directions then it wasn't differential because you would have two tangent points pointing in opposite ways I'm at a single point right so if at a single point the bottom of exactly that parabola the bottom of exactly this parabola there's infinitely many steepest way up notice that I said if it's non-zero then we can do this if it's zero there is no unique steepest way up also if there is no gradient then maybe there's multiple steepest way ups so here there are two at the bottom here there's exactly two there's this way and this way and they're both best if I take instead of this function I take let's call it p x squared plus y squared then at zero the gradient of f is zero and every direction is the best way up because it's round but once I get away from zero there's a unique way if you think about the tact there's that the thing where at the north pole all directions are south you can't go if you're standing at the north pole no matter which way you move it's south there is no unique north there is no east or west at the north pole because always are the way a maximal increase in south there's a similar statement here because the gradient is in the other direction so it's very important that we have a non-zero gradient vector otherwise we might have multiple ways that are steepest okay use this now some kind of a chain rule part of our goal which I won't get to next time to do the general chain rule but let's do a chain rule in kind of a stupid case for example suppose that I have some function let me draw it by a picture taking r3 of t so this is some curve into space so you can imagine that this is like a path well this is a path you're going to follow is say t t squared give me some path in space and we can imagine that will take a point in space and give me a number this is a path I'm going to traverse and my g of xyz is, and I'll just use the one that I did 2xy plus yz 2xy plus yz I can't remember long enough to long discuss it oh no, that's the gradient I'm sorry x squared of y plus xyz and so the composition first I do f and then I plug those xyz into g this gives me a function just of t this will give me t squared times t squared so that will be a t to the fourth and here I get t times t squared times tq to be t to the sixth this is a very simple composition much more simple than either of the input functions I can easily compute let's call this h h prime of t is quite easily seen to be 4t cubed plus so it's easy to see that it's 4t cubed plus 6t 6t to the fifth but let's compute it another way let's try and compute this by a chain rule kind of thing so usually I mean the chain rule would say that we want to take the derivative of the outside function here which is g and evaluate it at the inside function and then multiply that times so the usual chain rule if I have too many letters here if I have a function r of s of t and I want to take d dt of that then that will be r prime evaluated at s of t times s prime of t so that's for a function of one variable here in a function of more than one variable we want to compute something similar which is going to be well the only kind of derivative that we have for a function g here from r3 to r is the gradient so we might try to take the gradient of g and evaluate it at f of t and then multiply it by the derivative of f of t well f of t here is a vector valued function so its derivative will be used f prime of t for the notation of that which is going to be a vector and the only way that we can yeah so that's what we're going to do so let me just turn the page here so we have let's just calculate this somewhere there we go so here we have the gradient of g is going to be the vector so g is x squared y plus xyz so the gradient of g we take the derivative with respect to x we get 2xy plus yz for the x derivative now we take the y derivative we get x squared plus xz and now we take the z derivative and if we evaluate the gradient of g at f of t which means we plug in x equals t y equals t squared z equals t cubed that will give us the vector so x equals t y equals t squared so that will be 2t cubed plus t squared times t cubed t to the fifth here for x is t so that's t squared and then xz will be t to the fourth and then xy is t cubed so that's the gradient of g evaluated at f of t and f prime of t is the vector 1 2t 3t squared and the only way that we know how to combine these two vectors to get a number out would be a dot product so we take the dot product here and we dot that with f prime of t and let's see what we get well we get 2t cubed plus t to the fifth that's from this component here plus 2t cubed plus 2t to the fifth from here and then 3t to the fifth here and then if we add everything up so that is 2 plus 2 is 4t cubed and then on the t to the fifth I have 1 plus 2 plus 3 which is in fact 6t to the fifth which is what we wanted this is h prime of t so that worked now so that should be a general principle that we want to do so let's check well it's not really check but let's confirm this so let's write that as a theorem so suppose that I have my function g which takes rn into r and this needs to be continuously differentiable on some domain so I have some open domain that g is defined on and I also have f which takes r into rn which is continuous well actually it needs to be differentiable differentiable on some interval ab so that's just to make sure that where we're looking everything is nice and we have and it also has to satisfy that the image of where we're looking lands in this u so f of t is in u for t in my interval where things are nice so okay so everything's just nice now and now let's let h of t be the composition g of f of t then h of t is differentiable h of t is differentiable on the interval in question with h prime of t equal just what it was in the example the gradient of g evaluated by the image of f dotted with the derivative the vector derivative of f so that's my theorem and so let's you can hear me so let's we already saw it in an example let's see how the proof works so the proof just works by writing down how stuff goes so the proof says that h prime of t well that's just definition the limit as h o I shouldn't have used h so I'm going to go back and I'm going to change all of my h's to capital h's I made the same mistake when I did it in class that's okay so there's my capital h h prime of t so that's my difference goes to 0 so that's just the standard difference quotient for the derivative which is the same thing let's now forget about h and write it in terms of f and g this is the limit as h goes to 0 of g of f of t plus my little offset minus g of f of t then divide it by the distance I am away so if it exists if it doesn't exist well then that's not the derivative so now we look at that and let's let just for notation I want to call this guy here some point y so this is a vector in Rn and that'll be f of t plus h I'm just going to call him y for a minute he really depends on h and let's let f of t and let's let x be f of t so I just have these two vectors x and y so here's this is in Rn or R3 here's f of t which I'm calling x and here's y f of t plus h just a little bit off and here's my u and my u where everything's nice they both are in there because that was the point and so now if I consider the segment between them then I can use the mean value theorem on this so I use the mean value theorem which we talked about before so I use the mean value theorem here to see that so everything's inside the domain where f is nice everything's nice in here it's continuously differentiable on there so the mean value theorem says that somewhere between here and here that that g of y minus g of x the difference here divided by the distance that they're between the distance this is y minus x let's write it as y minus x that this is exactly equal to now this is wrong sorry g of y minus g of x is exactly equal to the gradient of g evaluated at this point x no some point nearby x naught which is maybe here some point in the middle dotted with the vector from y to x okay so now we can write the same thing let's just divide both sides by h for the heck of it so g of y minus g of x divided by h is the same thing as the gradient of g at this point x naught dotted with y minus x divided by h I just divided both sides of the equation by h that's okay we can do that alright and so now we're almost there so here now let's just take the limit of both sides and let's rewrite it a little bit well this is certainly this is true then the limit as h goes to zero well first let's write it back in terms of h g of f of t plus h minus g of f of t because that's what x is divided by h is the same thing as the gradient of g x naught dotted with f of t plus h minus f of t over h so that's what we have and now we can just take the limit of both sides as h goes to zero of all of this stuff this equals the limit as h goes to zero of this thing so this h here minus g of f of t over h this limit is the derivative of h if it exists here and again if it exists this limit well we can break it up into two pieces the limit as h goes to zero of the gradient of g at x naught times the limit as h goes to zero of f of t plus h minus f of t well this is since f is differentiable this is just f prime of t and here well since in our picture before x and y our point x naught lived on the line between f of t and f of t plus h is going to zero and since f is continuous as well as derivative differentiable but since it's continuous this is going to shrink to zero so that means this point x naught will converge to the point f of t so that will be just the gradient of g of f of t which is a vector so there we have it the derivative is just what it was in the example so that gives us the chain rule for this pretty much useless situation well it's not quite useless gives us the chain rule for this situation where we have some function h of t which is a composition of a function from where we have a function taking r into rn followed by a function taking rn back down to r so in this simple case we have we have that that the derivative works the chain rule is what we suspect it would be which is the derivative of the outside function evaluated at the inside times in this case is interpreted as a dot product the derivative on the inside function so we can use that to see that we can use that to see that if we have a function actually I think I didn't do this in class so I'm going to do this on Wednesday so this is a good place to stop so I'll stop there and so the two things that I will do on Wednesday which is in the next class is that I will do this in more generality and I will also prove that the gradient vector is perpendicular to a level set and so that is if I have some function f from rn to r it has level sets which are contours so or if I it has a contour here and that the gradient vector is always perpendicular to the contours and this is intuitively quite clear because the contours are the directions in which the function doesn't increase at all and if the function is nice then the gradient pointing in the direction of maximal increase will be of course getting away from the contours as quickly as you can that is moving perpendicular to the tangent of the contour so that's a good place to stop because that's where I stopped last time so I'll do that


This lecture is part of an online mathematics course on group theory and will be about the Jordan-Holder theorem So if we've got a group G We can break it up Into simple groups as follows. So we can find a chain of subgroups one is equal to G naught contained in G1 to contain in Gn By the way, I should say I'm considering all groups to be finite in this lecture such that GI is normal In GI plus one and GI plus one over GI is simple Meaning it's got no normal subgroups other than the whole group in itself And this is quite easy because if G is simple We're done if it doesn't we pick a normal subgroup to be one of these groups GI and then we just sort of continue Applying this to the normal subgroup and the quotient by the normal subgroup. So this is called a composition series and We can give some examples of composition series. So from the previous lecture We studied several groups of order 120 and these have composition series as follows first of all If we take the binary icosahedral group Then it has a composition series that looks like this and the composition factors Which mean these quotients well, this factor is obviously a group of order two and this group is the group of rotations of an icosahedron or the Quotiently the group a five Another group was the symmetric group s5 and this has a composition series which looks like this So this time this quotient is the group a five and this quotient is order two So you see we have two groups With two composition series and each composition series has a group of order two and the group of order a five only they're put together in different ways We can have some more like this for instance We can take the group z modulo 2z and this is contained in a five times z modulo 2z Or we can take one contained in a five Contained in a five times z modulo 2z and again this quotient here is order two And this is a five and this quotient here is a five and this quotient here is order two So we see that the composition factors don't determine the group here We've got two and a five as composition factors and here we've got two as a five and the group doesn't determine the order of the composition factors because in this group we can first do two then a five or do them in the other order for these two the order is fixed so There's a there's a lot of variation in what composition series can look like By the way, I should point out that each GI is normal in the next one This doesn't imply that G1 says normal in GN for instance if we look at one contained in Z over 2z contained in Z over 2z squared which is contained in the tetrahedral group a four Then each of these subgroups is normal in the next one But Z modulo 2z is not normal in in a four It's an example of something called a subnormal subgroup if you've got a chain of subgroups such that each is normal in the next Then we say that the one at the bottom is subnormal in the top So So there's a lot of variation in the composition series However, you see in this composition series, although the two and the a five occur in a different order They each both occur once and this is illustrates the Jordan-Holder theorem so these were two 19th century group theorists, which says that any two composition series of a group G Have the same number of each simple factor and We're going to prove this theorem by sort of driving a taxi cab around in a in a rectangular array So the idea is as follows supposed to have got two composition series So one is equal to a naught contains and contains an a m contains and g equals G with a i plus one over a i simple and We take one is equal to b naught contained in b1 Contained in bn equals g with each of the bi plus one over bi simple And we want to show these quotients are the same as these quotients possibly in a different order so what we do is we form this big array we take a M intersection bn, which is just equal to g and then we have a M intersection bn minus one go all the way down to a m Intercept b naught and then we have a naught intersect b naught here This is just one of course, and this is just the whole group G and Then we have a naught. So a 1 intersect b naught And so on so so we have this sort of two-dimensional array of groups wherever I got to this should be a naught intersect bn and We notice that each of these groups is a normal subgroup of the two groups above above it So what we have is this sort of two-dimensional array of groups as a sort of think of this as a rather unimaginative if you planned town where each of these vertices is some group and each of these lines is a quotient which is so which is a quotient of two groups and Each quotient is either a simple group or or one so so line is simple or Or one so it's so it's quite common for for these two groups here to be the same for example in this diagram here all these groups on the left hand column are the Trivial group so they're all the same as each other and we can imagine a taxi cab wandering through this town and The taxi cab is going to go from the this corner to this corner. So for instance, it could go along there And as the taxi cab goes along we look at the different Quotients it passes through so this taxi cab is going to pass through one one one one And then it's going to pass through these quotients b1 over b naught b2 over b1 and so on So this taxi cab route gives us the second composition series on the other hand we could also drive like this and this would give us Quotients one one one one and then we would get a one over a nought and so on and We can have other routes for instance the taxi cab could do something like this and We would again get some other quotients Along this route and what we want to do is to show that every taxi cab route has the same quotients Possibly in a different order and that would be enough to show that These two composition series have the same factors So how can we do this? Well, you notice that we can get from any taxi cab route to another one just by Repeatedly flipping it on some squares. So if we look at something like this We can have two taxi cab routes. So it might come along like this and go like that And we might have another one Which goes Like this so it's just the same except we've changed this square here and Obviously by repeating this operation we can get from any taxi cab route to any other taxi cab route We're assuming the taxi cab driver is honest and goes by the shortest route and doesn't stop going around in circles or something Which wouldn't make a whole lot of sense So let's look at what happens here well What we're getting is a group a i into sec b j a I minus one intersect b j a I Intersect b j minus one And a i minus one intersect b j minus one and We may as well quotient out by this group here since it's a normal subgroup of everything so what we have is is a group x and Certain subgroups a and b Which intersect in the identity? So we've got a particularly simple arrangement and as before each of these quotients is either simple Or is the identity and now let's look at what happens Well, if a equals b Then we can look at what the quotients are Well the quotients here are just a And the quotients here are just x over a So what happens if we go by these two different routes? Well, if you go this way We get x and x over a and if we go this way We again get a and x over a so the factors we get don't change what happens if a is not equal to b Well a and b a simple normal subgroups of x From which it follows easily that x must actually be equal to a times b and what we're getting now Is this quotient is a this quotient is b this quotient is b and this quotient is a So if we look at these two routes See this route we do a followed by b and this route We do b followed by a so Every time we change the taxi cab route We either keep the order of two factors the same or we switch them so so any two taxi cab routes have the same factors In possibly different order So this proves the Jordan holder theorem Which says that a number of times any given simple group occurs in the composition series doesn't depend on the composition series so By the way, this also works for groups with Operators That means it works for groups with some extra automorphisms on the groups and the main example of this is modules over a ring So we can think of a module over a ring as being a group with various operators So what this says is if you've got a module over a ring It can be broken up into simple modules and the number of times each simple module occurs is independent of how you break it up So This kind of in some ways reduces the study of groups to the study of simple groups Well, not entirely because as we've seen if if you've got various simple groups There can be many ways to stick them together into a group But some problems can be reduced to the problem of Solving simple groups. So now we want to know how do we classify Simple groups Well, the simple finite groups have been classified. There are 18 infinite series and two typical examples of this are the groups alternating groups a n for n greater than or equal to five and the Projectors special linear groups n over fq. This is provided n is greater than or equal to two and n is greater than or equal to three if Q is two or three and Various other series and then there are 26 sporadic groups Which don't fit into this classification. So the smallest is m11 which has about 7 9 2 0 elements and the biggest is the monster Which has about 10 to the 54 elements The proof of this is Maybe the most difficult Published proof in mathematics. There are various Computer proofs which are a good deal longer where a computer checks a gazillion different cases but as far as journal pages is concerned the classification of Simple groups is the most difficult and longest theorem. How long it is nobody really knows I've seen estimates of about 10,000 pages Or 20,000 or something that the trouble is this proof is scattered around so many journal pages that nobody really Knows quite how long it is If you want to see a summary of it There's a nice summary of it in these three books. There's a book by David Gorinstein the finite simple groups and introduction to their classification and then he had Follow-up he was going to publish three volumes on this so he had classification for non characteristic to type Unfortunately, he died before he could finish this but ash back a lay in Smith and Solomon finished it for him and Did the ones of characteristic to type so these three volumes? Total more than a thousand pages and they basically just give a summary of the proof without giving the detailed proofs so that the proof itself is I think there are very very few people who know most of it So how does the proof work? Well, you can't really summarize a 10,000 page proof in a sentence But I'm going to try it anyway. It works by looking at the central isers of an involution So this is the key theme in the whole proof so you remember an involution is an element of order to and Central is it is just the subgroup of things that commute with it and Brower pointed out that if you know the central is of an involution of a simple group Then you can determine it up to a finite number of possibilities So very roughly the idea of this proof is you pin down the possible structure of the central iser of an involution of a simple group And then given the central iser of an involution you try and identify the group In order for this to work a group actually has to have an involution If it doesn't then you can't really pin it down by looking at its the central iser and a group has an Involution if and only if it is even order so the very first step in the proof is to show that any simple group has even order and This is by itself one of the most difficult themes in mathematics It was originally proved by phyton Thompson in this rather famous paper solvability of groups of odd order So if a group of odd orders solvable then it can't be simple. So this shows that every Simple group has an involution and this single paper is it's about 250 pages long and it's Incredibly difficult to read in parts. So here's a typical page of it. You can see it Contains all these rather complicated generation of relation calculations I've no idea what's going on here I've tried reading parts of this and my experience is it takes me a good hour to understand each page of this paper So it would take 300 hours of hard work to understand proof completely Okay, that's all I want to say about simple groups for the moment So the next lecture will be studying outer automorphisms or more precisely the outer automorphisms of the symmetric group s6


Hey y'all, what's up? My name is Sonali. Welcome to my channel if you are new and if you are, definitely hit that subscribe button. And if you aren't new and you don't already have your post notifications on, go ahead and turn those on so you guys can get notified anytime I post a video. Today I'm doing a Sephora haul. As you guys probably know, the Sephora sale is currently going on. You can get up to 20% off anything in Sephora, which is a really good discount, especially on the higher priced items. I honestly don't remember the last time I bought makeup or skincare, so I thought I would kind of just go a little bit crazy and get everything new and just really like refresh everything that I had because I just felt like I was, you know, kind of like plateauing. Like my makeup looked like okay, but I knew it could just be so much better. I've been obsessed with like scrolling on TikTok and watching makeup tutorials and everyone's makeup is just so perfect and I was like, I need this. It's actually funny because I saw this one TikTok of a makeup artist doing this girl's makeup and I thought she looked so beautiful. So I was like, what do I need to look like this to the person that was helping me at Sephora? And she came in clutch. She gave me some skincare racks. She gave me makeup recommendations. I immediately came straight home, played with all my new makeup and skincare and I felt like a million bucks. So I want to not only share with you guys what I got today, but also share with you guys the tips that she told me. My skin is normal to dry, but obviously going to the winter months is going to be super freaking dry. So I definitely stocked up on some skincare that will really put the hydration back into my face. So I'm really excited about that. I guess we can start with skincare since that is like the foundation and I've realized that it is actually the most important thing when you're doing your makeup because I had this problem. Of my eyelids being so patchy when I put concealer on them and then I set them with my powder and they were just like really patchy and then my eyeshadow wouldn't go on smooth. And I was just like, how do these like people on YouTube have these perfect matte lids that literally look like a piece of paper that they can just blend eyeshadow on? I was like, how do they do that? And then the girl that was helping me told me that it's actually because my eyelids are dry and that I need to be moisturizing my eyelids.  So I started using that and it made the world of the difference. So all right, let's get into some skincare. Let's start with this one is the Youth to the People Superfood Firm and Brighton Serum. So it says it has kill tripeptide 5 vitamin C hyaluronic acid and I feel like I was taking a lot of recommendations from my sister, which she does have good recommendations a lot of the time, but her skin is very different from mine. She has acne and I don't really get acne as often as she does and her face is very oily. So I was trying to tell the person that was helping me, oh, like my sister recommended this and she was like, well, that's not really good for your skin. And it clicked in my head that our skin is very different. So I probably shouldn't be, you know, taking exact recommendations from her list, but it did help me with the brands I wanted to stick to. But anyway, so this product, she said, is good for just putting the hydration back into my face because I know a lot of like serums have like AHA and BHA. And she said that I guess I don't really need that because it's like a resurfacing thing for people who have acne. I don't know if I'm like spitting this out, right? Do you recommend this product because I have a lot of discoloration in my face and there are sometimes just like really uneven spots. It'll be like whiter right here and I just like don't know how and why.  So speaking of the moisturizer that I'm supposed to be putting on after that serum, I have moisturizers laying around in my house, but I kind of wanted to try something new since I was at Sephora and they obviously have the 20% off. So she recommended this. It is the Fresh Lotus Youth Preserve Moisturizer and I actually used this a while ago in college. And I guess I just like kind of like trying different brands so I didn't stick with it. But I have been using it for the past two days and I'm obsessed with it. First of all, I forgot how good it smells, but it also makes my skin feel so moisturized, but it's not thick. It's more on the gel side, I would say, but not even gel because sometimes gel just does not sink into my skin.  But this stuff is amazing, so I'm really glad that she really introduced me to it. I really don't get acne that often, but when I do, it's literally like the week before my period. I just get one random pimple and I just can't help but pick at it. I feel like everyone has this problem, but I'm just like, I'd rather have a scar than a bump, but I feel like I just need to get over the fact that a bump's on my face. So for the scars, I asked her what would be good for that. I wanted like a spot treatment kind of thing, so she gave me this. This is the Murad Invisis Scar Resurfacing Treatment. I've only used this one time so far and obviously this is a product that you will see the results kind of over time. So I'll definitely keep you guys updated about this, but I'm excited to try this out just because I feel like I don't have any spot treatments in my skincare collection. I purchased two things from Sephora literally the week before without knowing that the Sephora sale was coming. So I thought I would just, you know, kind of group them into this haul. This is the Laneige Sleeping Mask. I feel like this product right here is so well known. It usually comes in the pink package, but I thought this time I would try a different flavor. So this is a lemon sorbet and honestly it does not smell very overpowering. I feel like you can really only smell it in the packaging, but like when you put it on, it doesn't really smell like anything. So that's kind of nice because I didn't want it to be like too overpowering and give me like a headache or something because I do love Glossiersbomb.com, but I bought the Mango one and that one is just a little bit too overpowering. So that was it for skincare. Let's get into makeup. I'm so excited to show you guys this new foundation I tried out. For the past, I want to say like two years I've been using a drugstore foundation. It is the L'Oreal Pro Glow and honestly it is such a good foundation, especially for being at the drugstore maybe like $12 to $14 depending on where you buy it. But of course, bringing it back to TikTok, I'm watching all these girls with the most perfect matte skin and I was just like, I feel like my makeup just doesn't look like that and I want it to be perfect. So I decided to pick up the Airbrush Flawless Foundation from Charlotte Tilbury and you'll notice that I have a lot of stuff from Charlotte Tilbury because I feel like it's just all over TikTok nowadays. I don't know if it's just on my For You page, but I'm just seeing it everywhere. I honestly can't remember the last time I bought a foundation at Sephora or just like not a drugstore foundation. But I would probably pay $100 for this because it is that good. It is a matte foundation so she recommended to me to put like not a matte primer so nothing like benefits poor primer but I have been using like a Laura Mercier kind of like a jelly primer and just like a moisturizing primer and it works so well together. So this will be my new foundation favorite and you guys will be seeing it in the vlogs. I just feel like it makes my skin look literally perfect. Ryan actually walked in after I was done with my makeup the first time I put it on and he was like, you literally look like a porcelain doll. I was like, yeah, I know like it was so worth the money and I'm so excited that I freaking found this thing. And then getting into concealer, I just wanted to try something new. I have been using the Tarte Shape Tape since I don't even know when like so long. I don't even know did I ever use a different concealer. So I kind of wanted to try something new and I love hearing Sephora employees recommendations. I just love learning about skincare. I like hearing what they have to say about the products and just like learning what's in my products at home. So I decided to take her recommendation and I got the Too Faced Born This Way Multi-U Sculpting Concealer. I honestly thought it was a foundation when she brought it out because it's so thick. I feel like this is definitely going to last me so long. It says it's super coverage and I got it in the color Golden Beige and I forgot to tell you guys this color. This is in Cool Freud number 12 for the Charlotte Tilbury foundation. So hopefully that helps you guys if you're like my skin color and you want to order it online. But it is really helpful to go in the store and just like have them color match you. I never do it by myself because I don't trust myself. But I did use this concealer obviously for the past three days and I really like it. It's definitely thicker than Tarte Shape Tape. This one is another product that Tiktok made me buy. This is the Charlotte Tilbury Airbrush Ballist Finish. And I'm pretty sure the Tiktok I'm referring to is a makeup artist that works for Charlotte Tilbury. So obviously a lot of products was from that brand. When I saw this girl's makeup, I was like shook. Give me everything, add to cart, like put everything in my basket. I've been using the Laura Mercier Translucent Powder in Honey for a really long time. And honestly, it's amazing. I love it for a loose powder, but this is actually a pressed powder. And so I wanted to try it out. I thought it would be cool. And literally I swatched it in Sephora, put it on the back of my hand and I was like, wait, what? Like my pores disappeared. And it's honestly perfect, especially if you need to just touch up during the day. I kind of wanted to buy two. I know it sounds crazy, but I just wanted to like have one at home and then one if I needed to take it to go. But unfortunately right now I don't have that kind of money. So maybe later on, but seriously try this baby out. I don't think you'll regret it. I swear this is the last product from Charlotte Tilbury, but this is the Airbrush Ballist Setting Spray. This is actually the product that I bought a week prior to the sale because I saw Tiktok about it. And literally she swatched eye shadows on her hand. She sprayed this on and the eye shadows like did not move. And I was like, I need a setting spray because I feel like I've been just using kind of like refresher spray. Those kind of sprays are really good for just kind of melting your makeup into your skin, but not setting it. And I need that because I have orange makeup all over my face.   Let me know if you guys have tried this because I feel like this is like the new big thing. Hello, Future Sonali popping in because I am dumb and I'm literally missed out on two whole products. So coming back in the video and showing you guys these two, they're actually both from the brand Lawless. Let's start with this one. It is an eyeshadow palette. It's called The One and the colors in here are beautiful. It honestly really reminds me of my Sephora Pro palette, which I really like, but unfortunately I can't travel with it because the top of it like, I don't know how, but it broke off. So I decided to pick up a new eyeshadow palette because I haven't done it in a while and I just really vibed with these colors and the mattes in this palette. I feel like a lot of palettes these days have so many shimmers in them and I'm more of a matte girl. So I decided to pick this one up and then I got this bronzer from Lawless. I have been using the Huda Beauty Tantor contour cream, which I absolutely love and I still love it. I'll probably still use it. But honestly, going into the winter months, I just want to look bronzed and not just like very contoured. I don't know. It's a very different look. It's like a summery, like, sun-kissed look. So the girl at Sephora recommended me this and I'm in the color Blazed. It's just a velvet matte bronzer and there's no shimmers in it, which I really like because a lot of those sun-kissed looking bronzers have those tiny shimmers and I just don't like that. Alright, back to the normal video. The last two items I got are lip products and I honestly did not need any more lip products but I was like, why not, you know? So I got this lip liner from Makeup Forever. I'm actually wearing it right now. You can't really see it because I didn't put too much on but it is 608 Limitless Brown and it is a cool-toned brown. Honestly, on my lips it does look a little on the purple side. I'm not sure if it's because I mixed it with this gloss. This gloss is from Sephora Collection and it is in the color Hashtag Blessed 110 and I feel like it's just a little too pinky but I really like the formula so I might get another color on this gloss. But back to this pencil, it can actually also be used as an eyeliner and I'm really into brown eyeliners. So she definitely sold me on it because it's like two birds with a one-stone kind of thing. That wraps up this Sephora V.I.B haul. I hope you guys got some inspiration on things that you can pick up during the sale. Again, the highest discount you can get is 20% off on your V.I.B Rouge and if you're only an insider and you want the 20% off, just ask around to friends because I bet you can find someone with the code. To be honest, I haven't shopped at Sephora in so long and I'm not even a V.I.B Rouge and that is where having an older sister comes in. She gave me her code so Nisha, you came in clutch. Thank you. If you guys want any other recommendations, go ahead over to my Instagram. I have a whole story highlight of products that I've been using for years, hair care, skincare, makeup. So definitely check that out. All the swipe up links are attached to those stories and I'll also have all the links that I've talked about in this video so you guys can shop those links. I feel like I don't say this enough but the links down below are always pretty much affiliate links and I do make a little bit of commission off those links so it really helps me out when you guys purchase from those links and for all of those people that already do and already have purchased through my links, I appreciate you so much. So I will see y'all in my next video. Bye.


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the Braves. Oh. Refractor protector to 10, and it is gold. And it is Adley Rochman. Oh my God, it's one of 10 Baltimore. You saw that, Candler? Pick it up with a Merlin and a Wavefoot Club, Stadium Club. And with them with a couple of Breaks. Eric with the Mets and Cavalier. One as well. Actually, I'm gonna go grab Kenneth's other Breaks really quick. Yeah, it's pretty nasty. It's pretty nasty, I have to say, Candler. Evan, I gotta refund you for the NT hobby. I apologize on that. Like, check? Check. Check, check, check. Yeah, what's up, Brew? 23 NT Collegiate, yeah, we're out of them. Sorry about that. Oh. Let me go grab these next couple really quick as well because I'm just gonna do, I'm gonna do both of yours at the same time. Two TC Black Baseball, one TC Star Wars, one Lafa Stadium Club, and one Merlin. All right, one second again, everybody. And thank God, see you later, Steelers. Goodbye. P.J. as well, by the way. Yes, right after this, you will be next run, right after Kenneth. Perfect, Evan. What's up, Hooli? Some Eagles, I know. All done, man. That was rough, dude, that was a tough watch. All right, let me just double-check it. One Star Wars Black, one Tier 1, and then two TC Baseball, one more TC Black Star Wars. All right, good luck here, Kenneth. Let's do it, buddy. Going on, everyone. Sam here with Lance Forge Cards, we're in five boxes for Kenneth M. We got two 23 Top Star Wars Chrome Blacks, two 23 Top's Chrome Black Baseballs, and one Tier 1 Baseball. So good luck here, Kenneth. Let's do it, baby. Let's do your Star Wars first. Got Ron coming up next, followed by Evan, and then Jacob. See it. Peerle Royal Guard, Kylo Ren and Ray on the Fantastic Fights in a Yaddle. Otto is gonna be Groff Ditcher, that is Heath McGew. Oh yeah, plenty of Star Wars cards. Viennes. I have to disagree, I'm a Ravens fan Viennes, so I hope we don't have a rough one on Saturday. I hope you guys have a rough one on Saturday. Be fair. Look, Texans. Oh, chill. It's Figurin' Dan, 299, a Supreme Leader Snoke. It's a 50, Migs Mayfield. Nice one there, 35, 3450. Bill Burr. Get this guy out of here. An Otto Braille Organa. Let's go, baby. Let's go Texans. All right. Thanks, brother. Appreciate it. Two TC Black Baseballs. I'm rooting for the Texans, because Sam's rooting for the Ravens. He's gotta be a spellbag. I think the bills are gonna win. Bobby Whit, the blue to 75, Tim Anderson. Nice one there, White Sox, and Michael Bush. Did she say he had to be a scumbag? Yeah, he gotta be a scumbag, so. Masatake Yoshida, 3550 on the Gold Rookie Auto. Congrats on that one there. Oh, this is the sweetest thing you could've ever said. Nice one there, Ken. Next TC Black Baseball. Sure, he's probably next in line if someone doesn't participate, I would think. CJ is. That is a Paul Goldschmidt variation, Riley Greene. Riley Greene. And Purple Minidom is the 150, Shane Bieber. And we've got A. Shate Langlear. School of Minidom is 3550. Nice one there, for you, Ken, for the A's. And then A, Tierra Uno. Tierra Uno. Baseball. What's up, RJ? What's up, RJ? All right, Carlos Delgado, it's a 400. I hope so, I'm hope so, Ron, I hope so, man. Feel about them bucks? I'm happy, man, I'm happy for Jake. 299, Trish and Casas, and Kyle. Congrats on that one there, Ken. And last auto is Carrie Wood, four of 249, that's sweet. It's your one talent. Remember him? Oh, yeah. Remember to do it for your break there, Ken. Thanks as always for the support, buddy. Get it right out to my man. Yes, thank you. Assume this anyway. Ah, that should be fun. And like a knob or something or a barrel. No, they're AFC North, Hoolie, AFC North. Ron, this is you, my friend. Let's see something cool, dude. No, you're good. No worries, Hoolie. No need to apologize, man. Yeah, AFC North for events. Good luck here, Ron. Let's do it, baby. Let's see it. What's going on, everyone? Sam here with late in sports car. Drip in one box of 23 tops tier one baseball hobby for Ron T. So good luck here, Ron. Let's do it. Yeah, absolutely, absolutely, Evan. Yeah, I'll go grab your other two after this. I got you. Sorry I had to refund you for that NT Collegiate. I looked all over. I think we're completely out of stock on that one. Come on, see something big out of this, Ron. They trade Turner to 400. Congrats on that one there. Pretty cool one to start stud, man, is a Justin Steele rookie auto to 299 for the Cubs. He killed it, man. He was like a Cy Young contender for a while. Justin Steele. And then a Mark Mulder auto as well to 249. Little throwback name there. Not too shabby as well. Oh, not gonna fly you down, unfortunately, Ron. But thank you for your support as always. Get him right out to my man. Well, I think we're trying to get him right now, Evan. I think, I know Rich is trying to get him. Trying to get him right now. So we should have more in stock soon. I have to imagine. We've ripped through literally everything we had in like three days. It was unreal, dude. We flew through it. Because you remember how much of that we had in the back, right? We're down to a case and a half right now. I know. And let me go grab your couple more boxes as well. Some choice draft. Some draft choice, I should say. That's what you're gonna, IMAX. IMAX Collegiate and 23 Prism Draftics Collegiate. All right, one sec. Let me go grab those real quick. Got Evan and Jacob. What time will we go live? Five o'clock. But here, I'll send you the, I'll send you, let me put the, there we go. Here's the link for the break schedule. Five o'clock, Eastern. Will we go live in the morning? Yes, they will be on, I believe Fanatics. Let me see, hold on. Until six o'clock. I believe we'll be live on Fanatics, too. We live Fanatics tomorrow? Oh, Wednesday, that's what I meant. Yeah, Wednesday, the what? Like 10? 10 or 11 on Fanatics on Wednesday, RJ. And then the five o'clock stream on YouTube. What? It's best baseball and absolute football. All right, thank you, Ron. Appreciate that. Dude, I am so with you on that, Hooli. Drop, drop, pull in the streets. Yeah, no problem, RJ. Got you, buddy. I can't believe, dude, it's going, it's a blink of an eye, it's over. It's unreal.  Do you remember how much, like I literally remember us having the conversation, like I'm so ready for football to come back, man. I'm so ready for it. Yeah, and it's gone. And it's gone already. It sucks, dude. Sucks bad. It's like depressing. Yeah. That went all with August. All right, good luck here, Evan. Let's do it. No, it's tagged, perfect. Going on, everyone. Sam here with late in sports cards. We are ripping four boxes for Evan Y. We got one, 23 Panini Prism Collegiate Draft Picks football, 123 IMAC football, a collegiate football, and then 23-24 Panini Prism Collegiate Draft Picks choice basketball. So good luck here, Evan. We'll start with the football. Auto Raheem Blackshire 2199, red for Votek. Nice one there. Silver Jay Caner, Amika Amezi or NC State on the auto as well, base auto. Blue Wave, Kenny McIntosh, 249. Snake Skin, Charles White for USC. Bijan and Jalen Carter. That is Mustafa Johnson to 149 for Colorado. Yeah, oh yeah, we shipped to Europe. We shipped international once a month, really. All of our international goes out once a month. Silver Tim DeMoran, that is Zach Harrison to 99 on the Saturday Star Purple Ice Auto. Nice one there. We've got Red Tanner Morgan to 299. Blue Wave, Eli to 249. Will Anderson and Brian Rees. Silver Peyton Manning, that is an AT Perry to 75 on the Blue Auto, the New Orleans Saints. I'm just saving some money, I love it, Uli. Oh yeah, dude. Red Aaron Donald to 299. Red Wave is Bryce Young, not numbered. Lamar and Bryce Young again, the brilliant. Last pack here. Silver Eric Dickerson, Nesta Jade Silvera on the Hyper. Hyper Rookie Auto, Arizona State. Nice one there, Evan. Got Terry Bradshaw on the Mojo to 25 for Texas, or no, Louisiana Tech. Max Crosby, Red Wave. Johnny Manzell and Tony Dorsett. All right, we'll go into your two prism choices now. Good luck here, Evan. Good luck here, Evan. Red Scope, Sanogo for Yukon. The 88, Jared McCain on the Choice Exclusive. Some of Taylor Hendricks Rookie. Silver Jordan Hawkins Rookie. And Uzzelis Tubellis. Nice one there too, 40 on the Red Scope Auto for Arizona. Jalen Pickett, Dayman Lillard, and Keontae George. Second prism choice. Got Hakeem Elijah one. Timmy Allen and Jeremy Sohan on your Choice Exclusives. Red Wave, Derrick Whitehead to 80, 88. There's Jaker and Chris Paul to 88 as well. There he is. Silver Steph Curry. Silver Atosin of Bowman. And 240 is KJ Williams for LSU. All right, now on your IMAC Collegiate here, Evan. Good luck, buddy. 23, McAkelege, 23, McAkelege. Jason Spencer, we're coming. Let's go. It's Danny Dimes to 75. Sean's favorite quarterback. Well, now he's in the pennies range.  I hate to admit. He's the old man, buddy. 76 and 99, Bryce Young Tua on the Dual Jersey. Alabama Quarterbacks. Got six of 10, Tink Dell on the RPA. Nice one there, Tink Dell. 30 and 49, VJ Ojalari. Grats on that one there, Evan, LSU. Nice one here for Iowa State is a Xavier Hutchinson, 23 and 25, RPA. And really nice one here to end it. 14 of 25, Anthony Richardson. Congrats on that one, Evan. We'll get that into a mag for you, buddy. Very nice. Anthony Richardson, three-colored RPA. That'll do it for your break there. Evan, thanks again for the support. As always, buddy. Get him right off to my man. All right, we got Mr. Jacob and then Kenneth with another personal. We'll grab yours real quick, Kenneth, before we start Jacob's. All right, Jacob W. All right, Mr. Jacob, good luck here, buddy. Going on, everyone. Sam here with Layton Sports Cards. We've been three boxes, or two boxes for Jacob W. You got one, 23, a 22, 23, Topsy-Wave for Club Competitions, Merlin, Chrome Soccer Hobby. And one, 22, 23, Topsy-Wave for Club Competitions, Stadium Club, Chrome Soccer Hobby. Let's go, luck, Jacob. Let's do it, buddy. Start off with your 22, 23, Stadium Club. Ends up for Nannies for Chelsea. Goal force, Karim Benzema. That is a copy, Mano, 275. Echo's happening. Oh, now it's gone. Okay, good. Glad it's gone. Share and endure, refracta. That is a Valentin Carboni for Inter Milan. Refracta. You sold out until what?  Amir Dedek. Hmm. What do you got, bro? That is a Jude Bellingham. Roberto Carlos, Legends of Europe. Eden Desco. Refracta. Alfonso Davies. Odegaard and Morvez. Nicola Fazioli. That is a Pedri. Ocaio Saca. Frederico Ciesa. Ciesa. Takahiro Tomayasu to 350. That is a Bravo 6 Going Dark. Night vision of Eko Ekotike to 199. Bravo 6 Going Dark. That's fine, man. Really funny, though. And base, there you go. That'll do it for you. Jesus, that scared me, dude.  Had it. Had it. You know, Art of Goulash. Mon Luziata. Refracta, Auto Marc Casado for Barcelona. Nice one there. Rookie autograph. Casado. Refracta, Kevin De Bruyne. There's a Paul Vaughn. Blue to 75. Elise Ben Seguir for Monaco. Refracta, Virgil van Dijk, Liverpool. There's Wayne Rooney. An Atomic to 250. Nicola Borrella for Borrella. Inter Milan. That is a Patrick Kluverk. He's just out to me. Refracta of Magnus Acolloce. There's Francisco Conochiao. Ajax. There's a Paulo de Bala, True Green to 99. Or Aroma. Francie Perriot. Maccabi Haifa for Fracta. Dijon Kulusevski. Or Tottenham. Elise Ben Seguir as Monaco Rookie. Vintage Merlins, Chavi Simons for Eindhoven. Eindhoven. And Refracta Sabio for Eindhoven as well. That'll do for your break there, Mr. Jacob. Thanks again for your support, buddy. Get him right out to my man. Got Edgar picking up A. Select first off the line, WWE. Thank you, buddy. I'm gonna go grab that real quick. See you right at the box, everybody. Teen of 15. There we go. Yee-haw! Michael C. Yep, Vaughn. I am, we're going through the same emotions currently. Heart beating faster. Breathe. Oh my God. Look at that, on-card and everything. Come on, man. That's beautiful. Oh my God. Yeah. The eight of 75, Shohei Otani. That's huge. Finest Intimidators Auto, going to the ALS. Thomas K. That's huge. Holy hell, man. That is awesome. You get like... That is a Rookie Taj Bradley Frozen Fractor. Number three of five. Three of five Taj Bradley Frozen Fractor. That is an expose. Rookie Jordan Walker, Cardinals. Number to 50, nice SP. There we go. Good few boxes here. Another Yanks, one of one. Green Oswald Paraza. Nice. Congrats on that one, Yankees. Going to... If I could find him, that would be sick. Daniel C. Congrats, Daniel. Good stuff, dude. That's awesome. Big Poppy, Jersey Autograph. Five of 10 Red Sox. Heck yeah, boss, I love this one. That's Greg M. Greg has a lot of the Ortizas now, too. And... All right, Mr. Kenneth, coming up next. Ten Box Tuesday tomorrow with myself and Kyle. A lot of good mixes. Some regular old stuff as well. Good luck here, Kenneth. Going on, everyone. Sam here with the late sports cars. We're in one box of 23 Panini NT baseball hobby and one box of 23 Star Wars Chrome Black hobby for Kenneth M. Good luck, Kenneth. Star Wars Chrome Black hobby. Start us off. We got this, then Edgar. That'll probably do it for tonight. And we do have Personals again tomorrow with myself. That is Luminar Unduly to 199. Count Dooku and Anakin Skywalker and R2-D2. We pulled the card that at auction sold for 4.2 million. That was the most. Gorkorosh, nine of 25 on the Orange Auto. John Leguzama as Gorkorosh. Grats on that one, guys. That's a little orange on it. And one box of 23 NT baseball. Yeah, 101 Luca Logo Man. Yeah, that was the most. Got Redemption, rookie signature Jumbo material booklet. Drew Jones, 23, probably to 99 would be my guess. Nice one there on the Redemption. Oh, that's pretty sweet. Got A, Zack Neto, seven to 10 on the Jumbo patch for the Angels, like Neto. We've got A, Owen White, that is 20 to 25. Four colored patch, like two different shades of white. Owen White for the Rangers. Treasured Stats to 99 is Mr. Roberto Clementi, 48 of 99, 3,000 hits on the dot, rest in peace. Got the Pizza Man, Mike Piazza for the Padres, 13 to 25. Padres game use Piazza, can't be a lot of that out there. 97 to 99, Gabriel Hughes, die cut material signatures for the Rockies, the Grats on that one. Got 30 of 99 is a Roman Anthony, very cool one there. He is a stud. He trains with the holidays in the off season. Jackson and Ethan, smart move, very smart. The Red Sox, Red Sox again, this time it is gonna be a legendary signature, legacy signatures, 39, 49, Fred Lynn. Fred Lynn, empty definitive ink, Jose Canseco, 30 of 49. 30 of 49 for the AIDS. And then a 25 and 99, LED La Cruz, nice one there. Grats on that, Jersey Auto, absolute stud. Congrats, Mr. Ken. Gotta do it for your break there, Ken. Thanks again for the support, as always, buddy. Get him right out to my man. Okay, and one box for Edgar. Good luck Edgar, let's do it, buddy. Going on everyone, Sam here with late sports cards trippin' one box of 23, Pernini Select WWE Wrestling. First off the line, photo hobby for Edgar R. Let's go Edgar. Good luck here Edgar, photo. I love photos. I like photos. That is Gable Stevenson, 25 on the ringside. Good start, base, base, and Vir Mahan. Purple, Jerry Lawler, 299, and Hulk Hogan. Sava Vir Mahan, that is a Dexter Lumis on the relic card. Congrats on that there, Edgar. The Rob Van den. Sava Sami Zayn, Kurt Angle, and Odyssey Jones. Tri-color is Bianca Belair. Montez Ford and Lita. That is a Gunther on the choice, three of four. First off the line, exclusive. Nice one there, congrats, Edgar. First off the line, exclusive, three of four, Gunther.  Silver, Johnny Gargano. Got Matt Riddle, 2135 on the blue jersey auto. Or not jersey, but relic auto. And base. That is a Rick Steiner, 2149. Silver, Austin Theory, and Red, Topdalla, 2175. Base and base. One more hit, I believe in here. First up is a Blue Lita, 2199. Nice one there, Edgar. Base and base. Tri-color of Cora J, Ben Baylor, and base. Back here, die cut of Becky Lynch, and we've got Cowboy, Bob Orton. Nice one there on the Silver Auto. Very cool. And base, that's it for your break there. Edgar, thanks again for the support, buddy. Get him out to my man. All right, everybody. Thank you all again for hanging out today. Do appreciate it. Let me drop you guys a link to the break schedule tomorrow, 10 Box Tuesday with Kyle and myself. Again, personals will resume after. So thanks again for hanging out, guys. Do appreciate it. I will see you guys again tomorrow. So have a good one. Again, Bowman's Best Absolute will come out on Wednesday. So it's gonna be a good time. Good next couple of days. Have a good one.


Alan Hamill and Suzanne Somers are a couple who have not only made a mark in the entertainment industry but have also created an enduring love story that has captured the hearts of many. Before Suzanne's recent passing, the couple spent many, many years living in Palm Springs, an area they adored. Among their properties, they had a long time spot they recently had sold for $8.5 million. Final decision to sell came after Suzanne took a fall down the stairs, considering there were a hundred stairs here and she had to undergo neck surgery. Alan and Suzanne's former Palm Springs home was modeled after Lusto de Bominier in province France and featured glass walls that opened to scenic views of the Mount San Jacinto deserts. There are five buildings on the property, the main house and four villas. It offered a total of seven beds and ten baths. Some highlights included a 32-person dining room, a separate two-room kitchen, a 2,000 bottle wine cellar, his and hers closets, a safe room, and a stone guest house designed in the 1920s by Albert Frey. Alan Hamill and Suzanne Somers' relationship is a testament to power of love, resilience, and ability to overcome life's challenges together. Alan Hamill, born on June 30, 1936, is a Canadian TV personality, producer, and entertainer. Suzanne Somers, born October 16, 1946, was an American actress, author, businesswoman, and more. These two individuals, despite their different backgrounds and nationalities, found themselves drawn to each other in a way that transcended borders and any differences they had in their lives. The pair's paths first crossed in the late 1960s when they met on the set of the TV game show Anniversary Game. Alan was the host of the show and Suzanne was one of the contestants. Little did they know that this chance meeting would lay the foundation for a lifelong partnership. However, before Alan Hamill and Suzanne Somers became the power couple they're known as today, their relationship was marked by a tumultuous affair that tested their commitment to each other and eventually strengthened their love. When Alan and Suzanne crossed paths, it was during a particularly vulnerable period in their lives. Alan was already married and Suzanne, she was also in a troubled marriage. Their connection, though, was instant and undeniable, leading to a passionate and secret affair. This relationship wasn't without its challenges and difficulties and you might have not even known that about the couple. Pressure of hiding their fear from the public and dealing with their respective marriages put a significant strain on both Alan and Suzanne. There was a situation where societal norms and personal responsibilities clashed with their deep emotional connection. The turning point in their relationship came when they made the difficult decision to end their respective marriages. This marked a critical step in their journey towards building a life together, but it wasn't without its sheer criticism and judgment from the public and media. Couple faced scrutiny and negative press, but they remained committed to each other and their relationship. After divorcing their previous spouses, Alan and Suzanne made the brave choice to openly embrace their love. Their relationship transitioned from being this secretive affair, the public and enduring partnership. This period of their lives was characterized by resilience as they fought through adversity to be together. Your love story continued to evolve as they ventured into the entertainment industry together. Alan and Suzanne's shared passion for their work and mutual support became pillars of their relationship, helping them weather the challenges that came their way. Alan and Suzanne's romantic relationship blossomed and they decided to take the plunge into marriage in 1977. Suzanne was already a successful actress at this time, having gained fame for her role as Chrissy Snow in the popular sitcom Therese Company. Alan on the other hand was known for his work as a TV host and producer. Their combined talents and shared vision led to a series of successful TV projects. One of the most notable was the Suzanne Summer Show, which ran from 1980 to 1981. The show not only showcased Suzanne's charisma and talent, but also demonstrated the chemistry between the couple, which endeared them to their audiences. Over the years Alan and Suzanne weathered a ton of storms.  She bravely fought the disease and with Alan's supports emerged as an advocate for health and wellness. Together they co-authored several books on health and wellness too. Their journey through Suzanne's illness showcased their deep bond and the strength of their love. Alan was a pillar of support throughout her treatment and recovery and together they used their experience to raise awareness about alternative cancer treatments. Beyond their professional collaborations and shared commitment to health advocacy, Alan and Suzanne's relationship has thrived to their shared values, a strong sense of family and their ability to adapt to life's challenges. They had managed to keep their love alive and growing proving that love can indeed conquer all. Well, not only did Alan and Suzanne share a beautiful relationship, they also shared some beautiful homes. In 2021, Suzanne and Alan finally found a buyer for their long-time compounds in Palm Springs. They listed it for $8.5 million in January that year after attempting to sell it a few times before, once in 2013 and a few years even before that too. Suzanne and Alan's former village-like compound was set on a hillside in the Mesa neighborhood just south of downtown Palm Springs, California. This desert city is known for its hot springs, lux hotels, golf courses and spas. Suzanne and Alan purchased the legendary compound decades ago after they tied the knot as an escape from their main home in Venice, California. They happily lived here for over 40 years. Spending 28 acres of beautiful property as a unique layout with four individual villas and a main house. The mansion spent 8,826 square feet of space with seven bedrooms, nine full baths and two half baths. The spread is accessed by a private funicular that carries you up the hillside from the parking spaces below the main house and the insides are decorated both lavish and eccentric. Suzanne had a keen eye for interior design and she had personally decorated many parts of this house. The result was a blend of vibrant colors, opulent furnishings and artistic touches. Every room in the house told a story, flooding the couple's passion for art, style and life in the desert. The design inside the living spaces suit the desert backdrop with many stucco walls and stone floors underfoot. There are a mix of rustic and luxury accents like wood cabinets in the main house's spacious kitchen along with top-grade stainless appliances. The chef's kitchen further at a butler's pantry. The formal dining room here boasted high ceilings, opulent chandelier as well as some glamorous and vintage furnishings as well as room for 32 guests. This was suitable considering the couple often hosted gathering and events with their friends and family here. There was a main living room with wood beam ceilings and a wood-burning fireplace while elsewhere a sprawling wet bar. The main villa also contained Alan and Suzanne's one-time master suite, city view on suite bath and two offices. There's also a dramatic dressing room with sparkling chandelier overhead, his and hers walk-in closets and a safe room. Bonuses inside this structure include a 2,000-bottle wine cellar with tasting room, multiple private terraces and an outdoor shower. One of the multiple guest villas is called Rock House which is a stone-built structure set in the Rocky Mountain that rises steeply behind their property designed by legendary architect Albert Frey. The Rock House reportedly features a two-room guest suite with glass walls, a fireplace, hand-carved Balinese garden doors, a clawfoot tub, rock shower and peaceful meditation garden. Aside from the other guest villa as the property has a tented pavilion that overlooks the heated swimming pool and spa and a waterfall-fed pond. Among the many terraces to unwind and entertain, Suzanne's property even featured an amphitheater card from Rock Walls. Palm Springs Home reflected Alan and Suzanne's passion for wellness with a state-of-the-art home gym, yoga studio and wellness-focused amenities. After taking a peek at the most well-known estate of Alan, Hamilton, Suzanne's summers as well as diving into their inspiring love story, that'll wrap up today's video. Before we go, answer this question for me. If your spouse wasn't with you any longer, could you stay living in the same neighborhood or home? Or would you prefer a fresh start? Let me know in the comments down below and don't forget to hit subscribe and turn on your notifications so you never miss a video. Follow me on Instagram to chat and I'll see you all in another one. Bye!