In this video, I want to introduce the notion of a factor group or sometimes called a quotient group. You'll probably hear me say quotient group more often, although in Tom Judson's textbook, he typically calls them factor groups. It's very, very common, both terms. In my opinion, factor group is often introduced first in sort of like introductory abstract algebra courses like the video series you're watching right now. But as mathematicians mature, it tends to gravitate more towards quotient group. But again, maybe that's just my perspective. It's probably a colloquial thing. There's perhaps different regions of the world that use different terms. For me, being in the Intermountain West, quotient group seems to be the most common term when describing the structure we're gonna develop right now. So before we introduce a factor group or a quotient group, let me first introduce the idea of coset multiplication, which is what the factor group's operation is gonna be. Now, as a lima, to help us get to the, the coset multiplication, let's take a subset H of a group G. Let me, I don't wanna prove to you that if you take the product of H times H. So this is the product of sets, the so-called Frobenius product we've introduced earlier in this series. If you take the product of H and H, this gives you back H. So in terms of Frobenius multiplication, a subgroup is actually an idempotent element. Its square is equal to itself. Now, as these are sets, we wanna show two sets are equal to each other. We're gonna show their subsets of each other. That proves the inequality. So we're gonna first prove that H times H is a subgroup of a subset of H. That's what we're gonna first do. So take an arbitrary element of HH that we'll call it X. Well, if you belong to HH, that means there's gonna be elements little H and little H prime, such that X can factor as HH prime. That's what it means to be in the set HH. Everything in HH is a product of something with H with something with H, which those elements do not have to be the same thing. But H is a subgroup, right? So H is closed under multiplication. So if I take a product of two things in H, I'll get you something in H. So this thing belongs to H and that proves the first direction. HH is less than or equal to H. It's contained inside of H. The other direction is fairly simple because if I take something in H, then H can be factored as H times the identity. Because after all, again, it's a subgroup. It'll contain the identity in which case then H times the identity is equal to H. But that's also a product of things inside of H. So HE will be part of HH. And that gives us the other direction. You'll notice here that we assume that H was a subgroup. We use closure under multiplication. We use the inclusion of the identity. We actually don't need inferences to make this property hold. Just sort of as a little side that I just wanna throw out there that this equality held even if we didn't, even if the set's not closed under inverses, which of course it is true for subgroups. All right, so with that definition out of the, excuse me, that limit out of the way, let's now define the idea of a coset multiplication. So now let's suppose we have a normal subgroup. What do we know about normal subgroups again? Normal subgroups, we could define those as those subgroups closed under conjugation. So we could think of this way. So GXG inverse belongs to N for all X inside of N and for all G inside of G. That gives you, you could define that to be a normal subgroup. That's equivalent to be a normal subgroup. This condition of course tells you that GNG inverse is equal to N, right? It's closed under conjugation, which the way we originally defined it, which follows from this equation right here, we say a subgroup is normal if all of the left cosets are equal to their corresponding right cosets. So GN equals NG. That's the one we're gonna use right here. So we have a normal subgroup and take two arbitrary elements of the group G. Now consider the Frobenius products of the cosets AN with BN, all right? So again, what we mean by this is that when you take this AN and this BN, this Frobenius product, we're gonna take all the possible products of X and Y where X comes from AN and Y comes from BN, excuse me. So we're just taking all the possible combinations of these products from AN and BN, okay? So this is just the Frobenius product which we introduced previously. Now, one thing that's true about the Frobenius product is that it is associative. You can redo the parentheses and that doesn't change what the product is. And of course, when you see a single element, really you're thinking of it as a single 10, right? These are single 10s in terms of that Frobenius product. Now this is a statement that I made but we never actually proved for which if you wanna see a proof of it really quick, it's not too difficult of really an argument. So if we have sets like X, Y and Z which are subsets of a group, not necessarily subgroups, consider X times YZ as a set. What this is gonna look like is it's gonna look like all the possible elements of X times YZ, right? As X, well, let's back up a little bit. We'll call this one, we'll call this one W for the moment, right? All of the X times W's for which X is in X and W is in YZ, okay? But you can then equate that with all the elements X times YZ after all where X is in X, Y is in Y and Z is in Z. And the connection here of course is that what is the set YZ? It's just all the possible products of things from Y and Z, like so. So that's what X times YZ looks like. On the other hand, if you look at the set X, Y times Z, this is equal to, well, it's gonna look like W times Z where W belongs to X, Y and Z belongs to Z. But like we observed above, that W could be replaced with things of the form Y, X, Y times Z where X is in X, Y is in Y and Z is in Z, right? Cause after all, what is the set X, Y? It's all the possible products of things from X and things from Y. And since you're in a group, these things are associative. So element-wise, there's this correspondence between X, Y, Z and X, Y, Z. So that'll force equality of these sets. So the fermionist product is an associative operation. I just wanna make that very explicit here. So as I redo parentheses in situations like this, right, you redo the parentheses, there's no concern whatsoever. The Frobenius product is associative because the group product is associative. If the group was abelian, this would also prove that the Frobenius products are also abelian. The products of these sets is abelian. If that was the case, we don't need that right here. All right, so when you look at A N times B N, so this is a product of two cosets. I can redo parentheses, so I get a singleton A times the coset NB times the set N. Right, so now this is where normality comes into play. Because the subgroup is normal, we get that NB is the same thing as BN. I can commute those things around and there's no problem there whatsoever, okay? So that's, again, I'm gonna leave this on the screen. This is where normality is actually used. Then we redo parentheses again. This becomes ab times NN. And the previous limit we just showed that NN is the same thing as N right here. And so then we see that when we highlight what happened here, the product of two cosets is itself a coset, okay? So this actually defines a multiplication. So if we take the set G mod N, so the G slash N right there, it's often pronounced G mod N. That's the set of all left cosets, which for normal subgroups, there's no difference between left cosets and right cosets. So you could call it right cosets if you want to. And this turns out we actually have now a binary operation. I can take a left coset and I can times that by a left coset and this produces a left coset. So on the set of cosets, I now have a binary operation. What type of operation do we have here? Well, it's an operation that forms a group. And that's this theorem 10.1.12 right here. If you have a normal subgroup, then coset multiplication as defined above forms a group on the set of cosets. So the set of cosets itself is a group. We made a new group using a group in one of its normal subgroups. And this subgroup, this group is called the factor group or the quotient group.  Okay, so why is it a group? It has to be associative, has to have an identity, has to have inverses, okay? Associativity, I've already argued to you, right? The Frobenius product is associative as we just saw and coset multiplication is just a special case of Frobenius multiplication. So in the Frobenius product, we take any product of two subsets of a group. For the coset multiplication, we're just restricting our attention to only Frobenius products of cosets. That's an associative operation as we've already argued. This operation has an identity, okay? How do we know it has an identity? Well, the idea is the following. If you take EN and you times it by any coset, AN, right? Well, the rule from above gives you that you take EA times N, but as easy as the identity, you'll just get back AN. And since you have an associative operation, right? You have an associative operation if someone walks like an identity and quacks like an identity, it's gonna be the identity, right? It's gonna be the identity, it's gonna be unique, great. Same thing with inverses, right? I actually claim that the inverse of a coset GN is just gonna be the coset represented by its inverse. How do we know that? Well, take GN and times it by GN inverse, what happens? Well, by the coset multiplication, you're gonna get GN inverse N, which those become the identity and the identity is the identity coset is actually the identity of the quotient group right here. And so again, if it walks like an inverse and quacks like an inverse, then it's an inverse. So it has inverses as identities and it's associative, boom, it is a group. And that's what we call the quotient group, of course. So I want you to note that we've defined coset multiplication above using the subset multiplication, the Frobenius multiplication in a group and not based upon the representative of the coset, right? So like I mentioned earlier that since the Frobenius product is associative, we can redo parentheses. And so that's irrelevant of the representation. So when you talk about like AN and BN, right? You could choose different representatives. That would change this product AB, but I wouldn't change the coset ABN. Like the representatives could change, but the coset would still be the same. And so we've defined coset multiplication using the Frobenius product, which is well-defined, is irrelevant of the representation here, but it does require that be a normal subgroup in order for this equality to be validated. Now oftentimes, in like a first semester abstract algebra course, when they define coset multiplication, they don't introduce the Frobenius product and all of this intermediate stuff is missing and they're just like, oh, we're just gonna say that this coset times this coset is equal to this coset. And then they have to go through this long tedious argument that this multiplication is well-defined, that it doesn't depend on the representative. We of course, I mean, it's gonna take some effort to develop, but we took the approach that we defined at using the Frobenius multiplication for which no tedious, well-defined argument is necessary here because we've already established that, but we also have the benefit of we can use the Frobenius product or other things in addition to coset multiplication. So we don't have to prove that at all. Also another thing to be aware of is when you were talking about the quotient group, g mod n, the elements of the quotient group are cosets, right? Not elements of g. So g mod n is a set of sets, those sets being of course called cosets. So the elements of the factor group are not elements of the group. They're actually subsets of the group itself. So that distinction is important. Let's look at a couple of examples here. So we have to take groups with normal subgroups and then we can talk about a quotient group. So let's take the group S3 and let's take its normal subgroup, the alternating group A3. We've proven previously that the alternating group is normal inside of S3. The easiest way to see that it has index two, therefore it's normal. The cosets associated to the alternating group are first itself, one of the cosets is always the subgroup itself. Now, because you're index two, the other coset is actually gonna be everything else. So you have the even permutations and the odd permutations. And for S3, that is you're gonna take the identity with the three cycles and then the two cycles as well. Okay, those are gonna be our cosets. And so to form the factor group, these are then gonna be the two elements of the factor group, okay? So the factor group contains two elements for which I'm gonna show you the Cayley table with the coset multiplication over here. The alternating group is gonna serve as the identity element. If you take the alternating group times itself, you get back the alternating group. We saw that in our first lemma. If you take the alternating group times the other coset, according to coset multiplication, you take the identity times one two, which is one two. And this is gonna be the coset right here. And I want you to kind of verify that. If I take A3, if I take A3 and I times it by one two A3, right? This is a product of sets. So you're gonna take one, one two three, one three two.  And you're gonna times that by one two, one three, and two three. And so when you look at all the possible products right here, you're gonna get the identity distributed through, that's easy. So you're gonna get one two three, one three two.   So you end up getting this friend right here. When you do one two three with one three, you're gonna see that one goes to three, three goes to one, so one is fixed, and you end up getting a two three right here. And when you take one two three times two three, what you're gonna see happening there is one goes to two, and then two is gonna go to three, which then goes back to one. So you end up with this element right here. So you got the same three elements, it just showed up a second time. And now if you take one three two times one two, you get one goes to two, two goes to one, so one is fixed, and you're gonna see that one is fixed. So you're gonna get a two three right there. Then you take one two three times one three, one goes to three, three goes to two. So you're gonna get one goes to two, which is this one right here. And then finally you get one two three times two three. You're gonna get that two goes to three, but then three goes back to two. So two is fixed, and you get this element right here. All right, so when you take the product of a three with one two three a three, you get the three two cycles, and they actually show up three times. But you know, we don't really care about the multiplicity of such things. Well, I mean I do. I certainly as an algebraic commentatorist, I really care about the multiplicity of those things. But you know, we don't need to worry about that in this lecture right here. But the point is when we take the product of a three and one two a three, we get back one two a three. And then if you take one two a three times one two a three, you get back a three. And so this is a product, right? If I take this set right here and I square it, you're gonna get back these three elements and they're gonna show up with multiplicity three times. That can get a little bit tedious as you saw on that. The cool thing about coset multiplication is that you don't have to draw out all of the combinations.  So you get one times one is gonna give you one. You're gonna get one times one two because you want two. You just have to look at the representatives, right? One two times one two gives you one. And so that speeds up this calculation so quickly. Now, when you look at this Cayley table right here, this Cayley table looks a lot like the, well the cyclic group mod two. And that's because we see that a three mod a, sorry, excuse me, s three mod a three is actually isomorphic to z two. It's a cyclic group of order two. And in fact, this is something we see in more generality. If you take sn and you mod out an, you always end up with z two. One important thing to know about factor groups here is that if you have a group g mod n and you look at its order, well, the order is the number of elements inside the group, but this is this group of all the cosets. So the order is gonna be the number of cosets. The order of this group is gonna be the index of this normal subgroup. Well, since a n has index two, this group will always be order two. And there's only one group of order two, which is z two right there. Let's look at another example. Let's take this time the example. We're gonna take the integers and we're gonna look at the normal subgroup three z. So we're taking multiples of three. Well, this group has index three, right? There's three options, right?  So those numbers which are congruent to zero mod three, one mod three and two mod three, okay? And so when you start combining these things together, what do you get? Well, if you take a number, which is a multiple of three and you add it to multiple of three, you get a multiple of three. If you add it, if you take zero, that is when the remainder is zero and you add it to the remainder one, you're gonna get a remainder one. And when you take the remainder zero and combine it with a number which has remainder two when you divide by three, you're gonna get a remainder two, okay? And when you go through all of these things, right? So this here is our identity. We get zero plus three X right here. So this should just look like the row right above it, okay? If you take one and zero, you get one. If you take one and one, you get two. If you get one and two, you're gonna get three, a three plus a multiple of three.  And when you look again, go through all these combinations. This Cayley table right here is the exact same Cayley table as working addition mod three, up to relabeling of the symbols. And we in fact see that, that Z mod three Z is isomorphic to the cyclic group of order three. And this happens in greater generality, right? If you take Z mod NZ, this is equal to, this is isomorphic to the, this quotient group is isomorphic to the cyclic group mod N. Let's see, we'll do one more example and then we'll finish up the video there. So let's look at the dihedral group, right? The dihedral group Dn has a very natural normal subgroup that is the rotational subgroup. We mentioned that in the previous video that I mean, you could see that because it's index two. You can also see that every rotation, the only conjugates are itself and it's inverse. So the rotation subgroup is a union of consciousness classes. So it's gonna be normal, it's a normal subgroup. So we can mod out the normal, this normal subgroup. If we take the dihedral group and we mod out the rotations, well then we're gonna get back Z2, okay? Cause like again, we saw this above with the, with the symmetric group. If you mod out, this is subgroup, the rotation subgroup has index two. So when you mod it out, you have to get a group of order two. There's only one possibility, which is the cyclic group of order two. Now you'll notice that of course that this rotational subgroup R is itself a cyclic group of order N, it's isomorphic to N. And so sometimes in the dihedral group, you actually denote it as ZN or more commonly ZN without the blackboard font there. And so you'll see this, that DN mod out its cyclic subgroup turns out to be Z2 as well. And so this gives you some examples of computations involving these quotient groups. We'll of course do some more of these in the future. In the next chapter of Judson's textbook, chapter 11, we're gonna introduce the notion of a homomorphism, which will eventually lead to our, what's sometimes called the fundamental theorem of homomorphisms or what we'll call the first isomorphism theorem that tells you that there is a deep connection between the quotient groups we introduced in this video and homomorphism which we'll introduce in the next chapter.


The signal oil program The whistler that whistle is your signal for the signal oil program The whistler I am the whistler and I know many things for I walked by night I know many strange tales hidden in the hearts of men and women who have stepped into the shadows Yes, I know the nameless terrors of which they dare not speak Yes friends, it's time for the signal oil program the whistler Rated by independent research the most popular West Coast program In gasoline, you know it takes extra quality to go farther and signal is the famous go farther gasoline So look for the signal circle sign in yellow and black that identifies signal service stations from Canada to Mexico And now the whistler's strange story Safety in numbers It was a calm night almost too calm with a thick mist hanging low over the ocean In the distance the occasional wail of a fog horn on the Mexican coast a cold lonely sound Made Irene Thorpe glad his arms were around it thankful over and over again that there was someone in the world like Marty I'm lost Marty So lost forget it angel. It won't be like this forever. I tried to talk to Tom He won't listen to me. He will tonight What do you mean? He knows I'm aboard. I I Told you a charming husband. I had something to say to him. I don't know if you should have done that Marty He won't listen to me. Please Please Irene. Let me handle it. Will you don't know him? You don't know the cold steely way. He's Waiting look Irene, please Let's not talk about it darling Let's just think about us. All right money, but I'm afraid you do love me. Don't you?  Oh Listen angel get hold of yourself. I'll just see time not bad. It's something else What did you ever feel like? Did you ever feel you could kill with someone? baby Maybe I don't know what to matter with you tonight, but that kind of talk is no good. I know it isn't I can't help it Marty I want to kill him down inside. I want to kill him Not just you and me Everything that's happened during the past year all the secrecy the whisperings The way that awful Sam Romano watches me whenever I'm aboard Romano's a gambler all gamblers are suspicious Something's happening on this boat. I know it the money Marty. Why are they getting all that money? Can't customer 15 20 a night is not coming from there but couldn't Wait a minute.  He would like to see you in his office now. Oh I'll be right with you. Thank you. Thank you. I would not delay if I were you I'll be back in 10 minutes wait for me here, darling His wife is in love with you. I'll handle him baby. Hey, how about one more for luck? Oh Yeah, I'll be back in 10 minutes angel and from here on out we've got a lifetime together So he leaves you there Irene and strides down the deck towards your husband's office And the silence suddenly seems heavier the darkness blacker the fog horns lonelier 10 minutes past 15 20 you try to control your nerves try to tell yourself that Marty will handle Tom Without any trouble But as you glance at the watch on your wrist your hands shake So the figures blur and at the sound of the ship's bell You jump as if someone had reached out for you from the darkness 1130 Irene Marty has been gone more than a half hour.  He's made Without that cigarette. I'll do it. Yeah, but I said put it out There's no smoking on the ship, you know that Don't you think it's a little foolish thought I didn't ask you Kruger only a suggestion bud Customers might like a smoke once in a while and from the looks of the gambling rooms You need customers suppose you mind your own business. You better go to your room Irene I'm gonna see you later. We're gonna settle it right here. It's already settled I've ordered a boat for mr. Kruger my dear leaving in 10 minutes Now on Kruger you're barred from the ship. You'll come here if I want him to hear what I said Kruger You're off the ship for good Is it clear and what if it isn't? You're liable to wake up on the beach tomorrow morning covered with barnacles.  I've had all of this I can stand oh So you want to switch on to another track, huh? Well, you won't baby you're on this one to the end of the line Maybe I've got something to say about that you better go Marty. All right, but you're coming with me. She's not going anywhere Come on Irene what I said sure But I just don't agree.  No, don't do it Irene I mean Paper knife you you look at him on it. I don't know if he he He's dead Wait a minute. Wait a minute. We've got to think I'm free. We've got to get him out of here I'm free Irene out of it. Hold it yourself. You know what you've done is boys know. He's in here. How do we explain? Never mind that You know help me. We got to get him over the side. We'll tell him he fell You'll find the body No, no, I gotta take that chance. There's no other way to cover up Come on Comes right hand man Like the door so I can close it How to hold him here's what you do I stand in the corner by the body you're here We can't see me. We're having an argument you get it not another girl Now you do all the talking and don't let him inside that door.  What is it Andy I got to see the boss right away. It's important not now Andy Just can't wait the electric wiring is going bad down below. You can talk to him tomorrow I'll explain to listen. You don't know how important this is. He'd be sore at me I cannot explain to him. Do you understand bad? Okay, but tomorrow won't do I'll be back in 20 minutes Tell them it's about the wiring be sure. Yes. Yes. I'll I'll tell him good girl. We got by with it. He's coming back 20 minutes that time enough and we'll put the body over the side. Come on. Let's get this ball to open What's the matter? How the devil you work this? Marty you you know the combination don't know I don't nobody but his partner Sam Romano knows it. What do you mean? You mean we can't open it. I Rean everybody on the ship knows Tom's here in the office Oh my what are we going to do? Safety and numbers the signal oil company brings you another strange story by the Whistler on Last Monday's program. I read you a letter from a whistler fan who decided to test the mileage of signal gasoline for himself But if you recall after stopping at two signal stations, he was so impressed with the service He completely forgot to check the mileage Well tonight. I've selected a letter from another whistler fan. Mr. HT Fisher of Glendale, California, who writes I've been listening to your broadcast for quite some time and have enjoyed not only the stories But also your comments about signal service an experience. I had a few days ago proved to me what service really can mean About 10 miles out of India. Oh my car stalls. So I got a lift into town and looked up your signal dealer In a jiffy he drove out to where I was stalled located the trouble and I was on my way again This is the first time in 40 years of driving. I've ever been enthusiastic enough about service to write a letter But this signal dealer really gave me such fine service. I feel he deserves a pat on the back Well, mr. Fisher, we're mighty happy that it was a signal dealer who gave you the urge to write that kind letter And on behalf of signal oil company. I want to thank you Of course signal dealers being in business for themselves do have more incentive to do the kind of job that will make friends and keep them But it's letters like this which you whistler fans send in that make signal dealers strive to do an ever-finer job Of making today's cars run better and last longer Now back to the whistler. It all happened in a flash Didn't it Irene the feeling of foreboding that swept over you when Marty was called into your husband Tom's office The quarrel your hand on the knife and then Tom's body on the floor at your feet Somehow all through the past year you felt that the secrecy the Suspicions the whisperings aboard the gambling ship golden chance would lead to violence What you didn't quite think it would leave you in a predicament like this But you have to face it Irene Andy the ship's manager will be back in 20 minutes looking for Tom and Tom's partner Sam Romano Will be back tomorrow night to open the vault And you and Marty stand staring at each other knowing the body is low clock fast inside those three inches of steel Irene listen darling think you've got to think that combination must be written down somewhere catch you remember seeing no I can't Wait, what I remember. Oh A little yellow card in the safe at home a lot of good that does with Andy coming back in 20 minutes Oh Marty don't use Irene. We might as well What's that I'm calling Tom No, I bet I better Yes Waiting what oh all right. Thanks. I'll be right up Baby, there's our answer everything the boat's arm ordered for me Sure Sure, I'll wear his overcoat in his muffler As far as they know he's leaving the ship you carry the ball if we meet anybody mind that now You're a good actress a few minutes ago. You've got to do it again It's a tense few minutes. Isn't it Irene? Hurrying out to the boat deck Tom's overcoat up around Marty's face the hat pull low You make it to the water taxi and during that long ride to the mainland You're thankful it's a foggy night that the pilot isn't interested in his passengers The car's on the war for you left it and a half hour later The two of you walk up to the front door of Tom's big happy ender overlooking the bay Listen Andy I just dropped him off at the airport. I put him on a plane for Mexico City Mexico City You mean he went away without Hey listen you tell him about the electric wiring. Yes, yeah, I did he wanted you to take care of it He was in an awful hurry Something must have come up Andy. Yeah, like what? Well, I don't know He said something about some important business that Had to be there in the morning I can't help how it sounds Andy. I don't know everything about my husband's business. That's it. Mrs. Thorpe It's my job to know look for one thing he left the ball blocked makes a lot of dough He said you could bring the money here know what you're saying. That's about 40,000 bucks. I gotta pay off tonight I skip it skip it you wait right there. Mrs. Thorpe.  Well Irene You know, you can't go back to the gambling shift now. Can you? There's nothing to do but sit and wait for Andy stare at Marty light cigarettes put them out pour drinks and forget about them Stare at the little yellow slip of cardboard with a combination to the vault on it Until you can repeat them backwards and sideways and then finally at 4 a.m. All right, baby get rid of him fast. I'll wait the other room. We'll get back out to the ship as soon as he's gone I got it go on. Ah Sorry to lean on the bell like that. Mrs. Thorpe thought maybe you'd fall asleep. I am tired.  I hadn't noticed it until the night Sanity into you on the way home not much awfully tired. I guess and still he filed right on to a plane without getting a night's sleep, huh? Nothing makes sense. Does it I don't know Andy. I only told you what I know Just wondering what Sam Romano is gonna say when he finds his partner ran out What do you mean ran out something you wouldn't understand? Okay, mr. Thorpe is the cash orders is orders you're sure they're his orders, of course Walls ain't right over there. I left it open for you. Okay. I'll pull screwy though leaving 42,000 bucks in the sardine can like this. All right, Andy. Just close it and give the knob a whirl No one will know it's here. Just that there meet you here in the morning, Mrs. Thorpe So we can go down the bank together fine. I'll be there sure let's not kid ourselves Mrs. Thorpe It's stuff about a business trip in Mexico City the square is a three dollar bill Wonder what Romano is gonna think. Oh, well, I'm sure he'll understand Don't worry about the money. No, and don't you you see just to be sure I'm posting a few of the boys outside They'll stand guard for you, Mrs. Thorpe so no one can enter or leave the house without being seen night Andy yeah Nothing Yeah, yeah, yeah, we can't leave the house. We've got to wait. What about Romano? If we don't get to the ball before he does we will Look, I was thinking while Andy was here. We've got to slow down, baby. I've been going too fast We're safe as long as that body's locked in that vault I know I know We're on a spot. We're gonna make it You could have walked out on me money. No, baby I'm with you all the way Oh, kiss me money Oh Marty, maybe you should try to sneak out and get to this ship right now with Andy's watch those around the house Not a chance Sides, they'd never get a board without you Look darling just concentrate concentrate on playing that bank routine right tomorrow morning Rest we'll take care of when we come to it Oh Best go out to the ship as soon as possible the vault will have to be open for tonight's release. Yes, sir Well, I'd rather not See my husband won't be nice Romano ought to be here any minute. I'll get him to take care of it Romano What do you mean? Oh, did I tell you I got a little worried about your husband's business trip So I wired Romano he's getting in this morning instead of tonight, but I what's the matter Mrs. So I'm something wrong with that No No, excuse me. Andy. I have to go really what you're hurry. I'm sorry Andy I have to meet someone it's a personal matter Marty, I had to call. What's the matter? We can't wait till tonight. Romano's here now. Romano? Well, he wasn't coming. Can you he's here? Oh, we've got to go out to the ship right now I'll meet you at the dock in an hour. Wait a minute. We can't go out there and brought daylight They're boats going back and forth all the time with supplies and things the gambling deck will be closed up Marty Irene I know something happened to you. Go as fast as I could. We'll hop in I'll get the line Marty You will please turn off the motor and step onto the dock. Wait a minute. What's this all about? Please, senior You only implicate yourself further when you present arguments to the police Yeah I guess you were right about Romano We have discussed the matter at length with senior Romano That of course is why we must take you into custody I don't know what you're talking about. I am talking about your husband's unfortunate accident Now you will both come without trouble. Yes Yes, Irene You follow him quietly to the official car waiting near the dock Somehow there's no surprise. No shock You knew from the first from the moment you looked down at Tom lying dead on the floor of his office But you wouldn't be lucky enough to get out of this When you arrive at the police office, there's no brow beating. No questioning Just a series of mexican formalities before you and Marty are sent to separate cells to think it over The hours pass slowly agonizingly At seven another polite official brings you your dinner, but you send it back untouched lie down and try to sleep Somehow you manage to drop off for a few more hours When you open your eyes again the morning light is streaming in through the window And the same polite police lieutenant is opening the door to yourself We know it's this senora. What is it now? Did they bring you the newspapers? I know what they say I don't care anymore. Of course Perhaps then you would like to give us your side of this, uh, unfortunate affair We have spent the night discussing it with these Sunday fellow Also with your senor Krueger. I see All right, lieutenant. I'll give it to you All of it. I'm glad you feel that way senora You see the newspapers They might call it something else But to us it's murder The whistler will return in just a moment with the strange ending to tonight's story At the beginning of these programs, you've heard me say in gasoline it takes extra quality to go farther And of course if you've lived out west any length of time, you know signal is the famous go farther gasoline But perhaps you're one of those folks who like proof So i'd like to take just a moment now to explain what those qualities are that make signals good mileage possible Look at it this way If a gasoline gets extra efficiency from your motor it naturally helps you get extra mileage, right? Well when science increased the power of today's signal gasoline They also gave you quicker starting faster pickup and quieter higher anti knock And it's because of this improved performance that you now go farther than ever with signal gasoline That's why signal says check your speedometer for the best yardstick of gasoline quality You will find it does take extra quality to go farther And signal is the famous go farther gasoline Now back to the whistler Well, there's no use holding back is there Irene You can tell from the lieutenant's quiet confident manner that he knows everything It was a long chance right from the first From the terrible moment when you and marty put tom's body into that vault on the gambling ship And you know enough about gambling to realize a long shot like that comes through once in a lifetime So you've lost Irene Romano got there first and you lost your hands shakes as you light a cigarette Your voice trembles a little as you finish telling the lieutenant the story Well, that is most interesting senora There is so very much more in this case than we realized What do you mean? The others gave us a slightly different version Senora romano andi your friend kruger You know senora the sea holds many secrets It's unfortunate. You did not trouble to look at the newspapers It might have held yours. I don't understand I have a paper here. Oh, i'm sorry. I don't read stanish Oh I must explain them that my reference to murder was an indirect one I was speaking of the loss of 18 lives early yesterday morning aboard the golden chance What do you mean 18 lives? We believe naturally that as the wife of one of the owners you were a part of the enterprise The golden chance was a gambling ship in name only senora Its real purpose was to act as a transfer point for ammunition and high explosives For an insurrectionist group in central america. That's where the money came from money Is crazy. There's smoking I see it now Where's the ship? What happened to it? It seems they were a little careless about the electrical wiring senora There was an explosion on the golden chance. She sank without a trace Ship sold with tom's body everything went to the bottom I am sorry for you senora We pick you up on one charge Now we must prosecute on another Let that whistle be your signal for the signal oil program the whistler each monday at nine Brought to you by the signal oil company marketers of signal gasoline and motor oil and fine quality automotive accessories Signal has asked me to remind you to get the most driving pleasure drive its sensible speeds be courteous and obey traffic regulations It may save a life Possibly your own Featured in tonight's story were lorraine tuttle and gerald moore The whistler was produced by george w allen with story by brian thorn music by wilbur hatch And was transmitted to our troops overseas by the armed forces radio service This is marvin miller speaking for the signal oil company. This is cbs the columbia broadcasting system


Beth Ferris, thanks for joining us. My pleasure. Brookings Institution is obviously a very well-respected public policy organisation. What's your role there? I'm a senior fellow there, which means I do research on a range of topics. I also co-direct a project known as the Brookings LSE project on internal displacement that really looks at humanitarian issues, particularly internally displaced persons, which really brought me to the issue of natural disasters. You spoke yesterday on the issue of civil military response to natural disasters. Right. What's the key point you wanted to get across yesterday? I think, first of all, that the number and severity of sudden onset natural disasters is going to increase. We're going to see more mega disasters, more urban disasters, more places where disasters intersect with conflicts, where disasters intersect with technological failings such as in Japan. And in those situations, the military needs to be prepared to respond quickly and in collaboration with a whole range of civilian actors. It's a complex world out there. I think the military does some things very, very well. Other areas it needs to defer to civilian leadership. And what do you think is the key to making that work? I think knowing each other, but understanding different ways of working, different jargon, different language, different procedures. Military tends to work with a command and control structure, and yet most civilian humanitarian organisations are more consensus-building and process-oriented and talking through things. And sometimes those cultures don't automatically work together really well. I guess the traditional role of the military is a bit different, and they're being asked to adapt to a variety of situations. And yet some of the real strengths of military response are desperately needed in these crises, your ability to move quickly to the logistics, the lift power, the capacity, the communications. Those are all qualities that, as I said, are urgently needed in disasters. Beth, what do you think we've learned from recent disasters that you've had some involvement in? Well, this year particularly we've seen terrible disasters happening to developed countries. You know, the floods in Australia, the New Zealand earthquake, the earthquake tsunami, nuclear accident in Japan, tornado storms in the U.S., and I think we're learning that we actually have a lot in common with developing countries when it comes to response. You know, in all of those disasters, one of the big challenges was how to deal with all the offers of international assistance. Right. That isn't something that just happens in Haiti or Pakistan, but it happens in the U.S. and Japan as well, so there are some commonalities there. I think we've learned in the past year with the mega disasters of Haiti and Pakistan that the international system doesn't do very well in responding to more than one big disaster a year. You know, it's really overstretched when Haiti happened, you know, people were pulled out of other crises and people were moved from Haiti to work in Pakistan. The system as a whole just doesn't have the capacity to respond to lots of mega disasters. So you've, prior to Brookings, you've had about 20 years in humanitarian response. What drives you to continue in this field? What's your passion? I think it's just the most wonderful work in the world. I think it's such a privilege to be able to help in one way or another people who are suffering the worst ravages of war or disasters and so on. And in those situations, you always see this wonderful outpouring of generosity and creativity and compassion from local communities in ways that's truly inspiring, you know. It makes you want to go home and raise more money or write more articles or get more attention to those situations when you see the real dedication of people on the ground. It's wonderful work. I think everyone wants to help, everyone wants to know how they can contribute. I suppose the challenge is how do you pull those resources and coordinate them in an effective way. And I think a real challenge is that the whole humanitarian enterprise is driven by volunteers, by people wanting to give, by altruism, humanitarian ideals. And yet it's a world that's becoming, needs more professional responses.  You need to really work in a professional way. So that's sometimes a bit of attention. Beth, thanks for your time today. My pleasure. Thank you. It's wonderful to be here.


Chapter three of our book is about measures of central tendency, or really how to be concise and find a single number that can stand in for an entire distribution of numbers. It's an exercise in simplifying, of boiling things down to their essence. Now the reason we learn this, the reason we learn about these three measures which are the mode, the median, and the mean is because every single other thing we do in this class is based off of them you start by looking at what's the single most representative and then going off from there. So if we want to calculate variability or correlations or anything else we first have to know where the center of the distribution is. Also because if you think of Occam's razor and the simplest thing the center the single most representative number is usually the one you want to go with. Now in terms of what for, what does this matter to you? Well if you're selling products you might want to look at what's the most common one that people buy, that's the mode, or if you're a physical therapist you might be interested in knowing how many sessions do people need before 50% of them recover from their injury. That would be the median and for the mean you might want to look at what's the average income of people who become social workers in Utah County. That average, the standard one, is the mean. All of these can be useful. Now the important thing is in a nice symmetrical data set, I'm drawing a little bell curve here with my hands, those three numbers will all be the same but if you have a skewed distribution like most of the ones are low but a few go way high up and if you look at stuff like number of physical therapy appointments or the cost of houses really anything to do with money or hospitalization it's going to be skewed and in those cases you'll find those numbers can be very different. So understanding the difference between them and knowing how to choose one or the other in terms of what is most representative for your purposes is the major goal of this chapter.


Welcome to All Hands Update, I'm Petty Officer Ian Cotter. Women have served aboard ballistic missile submarines since 2011, and one officer shares her experience of that transition. Transition was very smooth. It took them a few weeks to get used to female voices on submarines, but I think that was one of the biggest differences. The Navy is also planning to integrate enlisted female sailors aboard multiple submarine platforms in the future. I think it will be a smooth transition just like ours was. The thing is, they're willing and able to do their job just as their male counterparts will be fine, just as the Navy, you know, all sailors are treated equal regardless of any diversity, including gender. Lieutenant Lavec is one of 18 female officers authorized to wear the submarine dolphins.


What is up guys? KarmaVedic here and welcome back to another dose. So I'm back in London finally after two amazing holidays in the month of January. First I went to Morocco with a bunch of friends from KCL Medical School. And now I just got back from Paris last night. Welcome to Paris day number two. So it's been a really crazy January. Today is going to be a day of getting my life back together, sitting down, writing down everything that I need to do. Think about what it is that I've missed and how I'm going to catch up on everything. Just basically organize my life. So the first thing on my list today is I have an appointment in order to get my national insurance number. These appointments are really hard to get and so I can't miss it, otherwise it's a mission to reschedule. So I'm going to go do that and then tackle everything that I need to do for medical school. Just when I was in Paris was just how efficient their metro is. You can actually open the door on the train while it's still moving. So there's no time wasted for people going in and out. On top of that, there was such easy transport between different locations, which isn't really the case here in London. I feel like a lot of times you have to move in sort of a triangle shape in order to get to your destination when a straight line would be so much easier. Anyways, I finally made it to the job office national insurance number building. Going to go in, have my appointment and catch you in a sec. All right, that wasn't so bad. All done with the meeting now and heading home. Passed by the store and got Alexia some flowers. She's feeling kind of stressed, so I wanted to get her a little pick-me-up. Anyways, it's drizzling slightly and I don't want to damage this camera and regret it later. So I'll catch up with you guys at home. Elevator's not working. I'm gonna sit down and continue organizing my life a little bit. Recently, I've been trying to use this planner over here. Alexia got it for me for Christmas. Whenever I use an online or digital planner, I never check it often enough to actually do everything that I write down and then that way things slip out of my mind and I forget things. So I'm trying to use a physical planner because that's what I use throughout all of high school and in my first degree at university and it worked very well for me. The most pressing thing that I have to do is tomorrow we have an interprofessional education pain symposium, which is basically where medical students, nurses, physicians' associates and other students in the healthcare industry work on a project together. We have to watch a case online and figure out what pain management and pain treatment we want to give them. Not gonna lie, there's a bunch of other stuff to do as well but just gonna start with that so I don't get overwhelmed. Every week at the beginning of the week, we get an email that sort of updates us on everything that's going on in medical school, upcoming deadlines and things like that. And so there's a bunch of things that have just been added to my plate. The first of which is an end of placement survey and on top of that, the stage three allocations are going live on the 10th of February. Next year when I go into my fourth year I'm gonna move into stage three. And in stage three you get placed in hospitals all over the country here in the UK and personally I don't wanna be really far away from London because that's where all of my friends and my family are and I'd really rather just stay where I am close to everywhere that I'm familiar with. So that's something I definitely need to think about over the coming weeks, decide where it is that I want as my first choice, my second choice, et cetera. Oh God. We've also got a medical ethics and law workshop coming up. Definitely need to write a couple of things down in my journal. Gonna quickly fill out that survey and then go back to preparing for the pain symposium tomorrow. All right, all done with the survey. Looks like for the pain symposium tomorrow there's a couple of lectures I need to watch and a case study that I need to go through. So I'm gonna grab my iPad. And without further ado, let's just get to work. All right guys, as stressed as I am about the remaining work for the day, I'm really hungry and I think I should probably before just pushing through all this work, it's not gonna help me in any way. Alex, he and I are gonna keep it very simple. We're just gonna make some fish fingers with rice. No, we would usually make, but we just need a quick meal because we're both really busy today. And then we need to get back to work. Gonna make a quick coffee because I'll learn this levels are and then back to work. It is time, it is time for another study sesh. The last thing I wanna do right now is sit down and have a four hour study sesh, but there's too much to do. It has to be done. Wait a minute, something's missing. All right, so like I said before, I'm still working on that pain symposium thing that we're gonna have tomorrow. I've got another lecture and then the case study to go through. So hopefully after that we can move on to other things. So enjoy this super fast forwarded version of me going through a bunch of work. Guys, I'm all done now. That took way longer than expected. So much time. Anyways, I'm all ready for this pain symposium thing. I'm gonna cross it off of my list in my journal, which I'm now using in 2020. And then yeah, I still have a lot of like small admin things to sort out. I need to go through all of the emails that I've accumulated over the last week. I need to figure out where I am for my next clinical placement and psychiatric placement. Also, this semester I've opted to take part in an extracurricular activity, which is called signed up. And basically it's led by students, teachers and patients. And it's all about getting medical students like us to practice our history taking skills, our osky clinical skills. So yeah, I'm gonna have that once a week for I think the next eight weeks it was. So it's a good chance for me to make sure I'm on top of everything and you know, just getting things done. All right, so let me quickly find out where that is tomorrow and if there's anything that I need to do to prepare. Oh, first session takes place on Tuesday the 4th of February. Okay, great. That's in a week from now. The only slightly annoying thing is that it's held Denmark Hill Campus, which is decently far from here. It's like just over an hour of travel. So it's gonna be quite a bit out of my way and take up a decent amount of my time. But I think it will be worth it to get in that increased practice because oskies are something that you can't really learn from just reading a book. You have to actually practice and do it yourself. Also gonna leave a quick review for the Airbnb that we were staying at last week. In case you're wondering, it was excellent.  I think I get something like 10 to 15 emails a day between medical school and university related things and then the Karma Medic channel. Honestly, it takes so much time to go through them every single day and make sure that you're caught up. And I hate leaving emails for the next day because it just piles up infinitely. So I think I'm gonna get a head start on that. Go through as many emails as I can. I won't be able to finish everything because there's just too many but might as well make a dent in it. All right guys, I actually managed to get through quite a few emails. So really happy about that. I'm not gonna make a list of all of the lectures that I've missed over the last week. Just gonna make a list of those in my trusty notebook so that I know what it is that I have to do in the coming days. All right guys, you know what? I've actually managed to do quite a lot today. It was pretty productive. You know that feeling when you come back from holiday or you haven't been in school for a while then everything seems so unorganized. It seems like you have a hundred things to do and you don't know where to start. That's how I felt at the beginning of today. I just wrote everything down, went through it one by one and here I am done feeling a lot, a lot better. I hope you guys have enjoyed this productivity vlog of me getting my life together after a little bit of holiday. If you did enjoy, please leave me a thumbs up on this video and subscribe to the channel to see more content from me in the future. New medical school videos on this channel every single Thursday and I will see you in the next one. Peace.