Search for a home on any device, any time, anywhere. Private Property, a home for everyone. Happy Friday and good afternoon. Welcome to When Home on Afternoon. Express your favourite show to get you into the weekend. My name is Danilo Acquisto. Now as you know, South Africa's premier interior design reality competition is proudly brought to you by Private Property and each week we follow the journey of our three designed duos as they transform cluster homes at the Eye of Africa Goldford residential estate from white box spaces into lavish homes, one room at a time. Be sure to let us know what you think of their efforts on social media using the hashtag Win A Home and I love reading all of your tweets. It's been an exciting journey thus far so let's have a quick look back at what happened last week when the master bedrooms were revealed.  At the final countdown, yours truly did not get a warm welcome. This time, I'm not even allowed inside. That's right! But when all was said and done, the designed duos delivered dream-like spaces. Team House and Leisure were commended on their soft and restful retreat. Walking into this room, Team House and Leisure, I'm immediately transported to a place of just happiness. Team VC created a modern master bedroom that the judges loved and some would describe as sexy chic. Bed mirror, recipe for babies. Team Habitat wowed with their decadent yet contemporary room that rarely impressed guest judge David Muirhead. Versace on the dance floor. The designs were so good, there was almost a tie. But one designed duos rose above the rest. The winner of the master bedroom challenge is... Team VC! Now that all the designed duos have won a challenge, the heat is truly on in the competition. I think it's appropriate to say flames as our designed duos are definitely feeling the heat. Not only in the form of huge pressure, but it's also their proverbial wallets that are burning. Our designed duos have a cash budget of 300,000 rand each for the entire house. That's in addition to all the sponsored elements. And after the master bedroom challenge, Team Habitat has 114,000 rand remaining. Team VC has 104,000 rand remaining. And Team House and Leisure is in serious trouble with only 85,000 rand left for their cash budget. In terms of high-end interior design, that's not a lot of money. And our designed duos have three spaces left to decorate with an expensive one next. Hello contestants. It's time to get cooking as it's challenge time again. The next room is the heart of the home. It's a room that's required to be more functional than pretty as it serves an essential function in the home. It feeds the family. Your next challenge is the kitchen. I'm quite excited about doing a kitchen because it's like the central space of a home. The heart of the home. You know, so it's quite important. Very important. You've got to eat in a home. We don't want our guests to starve. No. So it's quite a big pressure and we hope to do our best. Both of us have experience in the kitchen industry. Paul works at a kitchen company and so did I. So we do know how it works, but time is not on our side. Yeah, because most of the time doing a kitchen requires like it's not a two weeks challenge. So pulling this, it's going to be quite amazing. You'll have just 10 days to design a multifunctional and breathtaking kitchen. Now I know you guys have been working really hard up until this point and I've been pretty harsh on you. So we had went home and I've decided to give you a teeny tiny cash bonus to reward you. I would say 5,000 rand sound added to your budget. Yes! How about an extra 10,000 rand? Yes! What happens if I told you I'm giving you 20,000 rand for your next challenge? I'm so excited that I fall off my chair. I just couldn't believe Lamont. I thought getting like 5,000 and it's like 20,000. I had to catch you. 20,000 rand! Well, of course I'm over the moon. I feel like I've just won the challenge. It's going to be a send via kids sign. Razzle, dazzle them and chop it up. Yes! 20 grand is just what we needed. Yep, we're going to use it and use it and use it some more. I also have another surprise for you. With the courtesy of Klicks, you'll also be getting an extra 10,000 rand voucher for anything you'll need from the Klicks stores to make your kitchens shine. Oh my god! Tanira was like Oprah. I mean, he was. You get her. You get her. You all get 30,000 rand. Now we're just balling. Now we're just balling. That is so super exciting. We can get all... We can save from our budget and use the 10,000 rand from Klicks to get accessories for the kitchen. And appliances because they're so expensive. On that note, contestants go forth and design and I hope you can handle the heat. In their new home, a housewarming gift is always welcome and thanks to our generous sponsors, the gifts just keep on coming for our design duos. All three teams get premier home appliances by Grundig for the kitchens and guess what that means? Shopping, shopping, shopping. We're excited to be in Grundig because it's German products and it's like having sports cars in your kitchen. Hi guys, I want to say welcome to our Grundig stand here at the Hirsch Four Ways. I'm so glad that you guys could join us today. I'm really excited that you guys have chosen Grundig to be your kitchen appliance partner and we're so excited to be a part of the show today. Just some facts about Grundig. We're in South Africa for about four years now. We're a German engineered brand which means high performance so you know that you can trust the products. We have a five year warranty on all the products which means that you have peace of mind for the home that you are designing so the person that wins that home knows that they have peace of mind on all of their products so that's really good. If you look at the design of the actual appliance you'll see that we have the piano key feature which is for the master class so that means whoever wins this home has the potential to become the master in the kitchen. The design is music to our ears. I mean they've designed it like piano keys. It's a piano key oven. Well how you press those buttons? It's like... Is it a thing? Machenachino. Or somebody.  Yes! Roxanne, we're not quite sure if we should take the bigger oven or the smaller oven but the thing is when it comes to baking we're quite concerned about the space.  The multifunction oven is a great oven to have but if you have a smaller space then the microwave oven is actually quite great. It can work independently as a microwave. Also they have 38 chefesses programs that means that you can look up recipes which include baking recipes. And with regards to your question about baking if you open up the oven you can easily remove these trays move your grill to the bottom and you can put in a beautiful cake in there. Yeah so basically this oven is nice and compact and it has you know the multifunction of the microwave and the actual oven so it's actually a two in one. So I don't know if that's going to help you make a decision a little bit more. Suppose these are hob options what do you think? I've been eyeing this guest on the last hob I think it looks beautiful and I think because we're going with like the multifunction kitchen since it has that classic feel to it this like the guest would work better with it. Yeah don't you think something more minimal and just something that disappears? Well I'm not all for minimalism but then I've you know me. I've always had like a thing for traditional looking guest stoves and I feel like with the stone it'll work perfectly so I don't know what you think. Yeah I think because our kitchen is quite modern I think this will be great. And I love that it has an option for using different sizes of pots and the fact that it has like a child log system. No we can totally go for it. I like it too. Finally he agrees. Ooh what the fridge! I gotta have it! Hi guys. Hello. So tell me what are you guys leading towards and what are your refrigeration needs? Sure. What we know is you're going to find a big fridge. So we sort this one and we like how it's separate like in terms of that's the fridge that's the freezer. Okay. And it's got a lot of freezer space. And which is the other fridge that you sort of leading towards as well? Then we looked at this one. Oh yes okay. But then is this one smaller than this one? Well guys it actually all depends on your needs. So basically this is a side by side so it's a combined fridge as well. So you have your fridge in one side and you have your freezer in one side but it's one unit. You've also got a water dispenser in the front and an ice maker. The best thing about all our fridges is that it's A double plus that means it's energy efficient. These two units are actually two separate units. So you have your freezer and then you have your fridge in one side. The best thing is that if you want to separate them depending on the space that you have in the actual home you can do that as well. Although I think it looks good as a pair. So all the essentials that make a kitchen are pretty functional. We were looking for those items. And they were fab items. Beautiful. And the bigger the better. We've got two fridges stand alone. Yes. Yes indeed. And the spending free was not over yet. Wow. Shopping at Clarex we want to fill up those kitchen cabinets. Absolutely. If our heart of a herd needs to be hearty full of things. Hi guys. Welcome to Clicks. Here at 10000 vouchers go and enjoy. Thank you very much. 10000 it's like. It's amazing. Like 10000 was just a bonus. Like we're going to buy closer with this one. Now we're just falling. Now we're just falling. Yes. Charter. Charter. Cambrook. We need an eye. We need an eye. We need a digital hand mixer. Charger. This is.  We need this because it looks exactly like our kitchen. And it's colour changing. Overruled. What about this one? Okay. You can have that. I think we want to see our water boil. Yes. Take both those toasters. You need both. Before this all comes back well. Sabotage. What's that? Sabotage. Sabotage. Sabotage. Sabotage. Sabotage. Sabotage. Sabotage. Well, since we've got the same Clicks amount with all the jurors, it's a competition after all. We might as well mix them as that. Ten thousand. So if you need to take motion that's off the floor so that you make sure your competition doesn't get it. You gotta do what you gotta do. I checked the weather focus and apparently that's how you can appear. I'm so excited. The smart life. Shelf. Super blender. I want the yogurt maker because I think whoever gets the house can help laugh. We're thinking it. We want that coffee maker. It's perfect. It's perfect because our client loves to drink coffee. And also wine. And wine and coffee. Yep. Clicks have got merchandise to die for. I mean every aisle is a winner. You want more, more. And it's got more, more. One of the typical Clicks is charge it. If we like it and you gotta have it, you just charge it. At Bulgatix, a clever app is helping Team VC make their flooring decision. This flow quiz was quite fun. The app from Bulgatix is really fun. I clicked, she clicked and then we clicked to become air. And as designers, I think we're very calm. I think that's why maybe we are the air element at Bulgatix. You're welcome. Did you do the quiz? Yes, we did. And you came out as? Air. That's what we came out as. It said air. Wonderful. Let me explain to you guys how the system works. What we did is we created this collection of products. We have different colors and different designs, different textures. But we realized that it's sometimes difficult for the consumer to make a decision. So we took the five elements of life, fire, earth, spirit, air and water. And we grouped different colors and different ranges together under them. And we then created a quiz that you would do. And when you go through the quiz, you would come out under one of these elements, either fire or earth or spirit or water. And you were asking specifically for something that could work in your kitchen environment. So Charleston, I would recommend, and as you came out in the quiz as air, there you can see the color box wood out of the Charleston range. Is that something that you would consider? That's perfect. And we also love the trains that we see with it. But we want to do like a herringbone technique. We just don't want to have like a plain floor. Because remember, the wall, the floor is our canvas. So if we do like a perfect floor, that's going to blend in nicely with our walls. We'll be very happy. Okay, I agree with you 100%. I think it's quite a nice new trend and fashion to go with the herringbone. It will definitely work. Will it work in the kitchen since it has a lot of moisture and a lot of movement? Definitely. This will work on the floor. It is not wood. So it is manufactured for the purpose of handling those water areas in the home and in the kitchen environment. So this floor, perfect for kitchen. Mipo, you asked about the herringbone. What I can recommend, for example, is that you combine two shades. So if you look at the lighter color that you are looking at and that falls under that element of air, you could add a darker gray to that. You think the two colors would work? We'll just have to think about it. Yeah. 100%. You can go and play those other options. I think it will work perfectly. Awesome. Decisions, decisions. From the appliances to the floors and everything in between, our design dios will have to consider every detail to create the right flavor for a winning kitchen and it doesn't stop there. Still to come on the show, our design dios explore the psychology of color. Stay tuned. Because if it doesn't improve people's lives, it isn't Plaskon. Plaskon, designed for life. Welcome back to Winner Home on Afternoon Express, exclusively on SABC 3 for a tad bit of design inspiration for your weekend. Now, did you know that of all the colors in the color spectrum, blue is an appetite suppressant.  This is all part of what's called color psychology and our design dios will have to choose colors very carefully if they want their kitchens to be inviting and there's no better person to advise them in this field than Plaskon's color expert, Claire Bond. We love Claire. We love Claire. And color is essential to any design. Crucial. Crucial. Welcome everybody. It's great to see you here at our paint master class. Now, the wonderful thing about color and color psychology is that each and every one of us reacts very differently to the colors that you're seeing all around me. So let's have a look at our warm colors. Peach. Orange and peach are fantastic for nurturing. So they're great colors. They've got vitality in the space. They give the people in the space the feeling that they're happy and they joyous. Red has the longest wavelength. So you've heard me say before that it's the most stimulating, the most energetic. It's an excellent color for a restaurant because it makes people order more, it makes people want to drink more and it actually makes people feel happy and they kind of want to dance in the space and enjoy the moment because of its very stimulating wavelength. The orange is quite a... it's a color of strong vitality and health and it really assists people with moving into challenges. Purple's not as stimulating as your red, orange and yellow but it's still a wonderful color for helping people communicate and it allows open communication in a social space. So I would use it in a lounge or that sort of thing where you actually want to bring people together and enhance communication. Yellow is a wonderful color where you want to bring joy into a space. So if you have someone in your family who suffers from a little bit of depression or they've got a bit of a melancholy personality, yellow stimulates the nervous system and it helps people feel more joy. So it really is a good color for any area. It's not just a color that's great in a kitchen or a bedroom. What's going through my mind regarding the color psychology class is the fact that warm colors are great for the kitchen, especially if you want people to be appetised and I think that's what you always want. Now let's move into the cool side of the color spectrum. So green is linked to the heart chakra. So that's all about balance in your life. So these days when we really have no connection to nature at all because we're always on our laptops or mobiles, this color comes highly recommended. Then we move into a much more sludgy, olive-y green. People are versed to green. Actually need to help others more. They're very independent and very goal-orientated and directed towards self. So it's important to introduce green into your life so that you actually give a little bit more. So blue is a calming color. It'll calm your energy down and it'll also make your space feel calmer and much bigger. The deeper blue you go gives a different resonance. So if you go for something like a cobalt or electric blue, it's much more stimulating than a cool blue. So just be careful of that. Paint effects are back and very popular. For me, beautiful translucent layers of color make a space look incredible. On top of those layered colors, I would suggest you either go and buy a ready-made stencil or even cut your own stencil because most of you are very artistic. Well, you're all artistic in this room today. Claire, when choosing colors for kitchens, what colors would you recommend to encourage an appetite? I would recommend the orange palettes. So as nurturing as the apricots are, the more saturated the color and the more orange, the more it allows people to feel hungry and want to eat delicious food. Color is another thing, especially when it comes to color psychology and other stuff. You need to know your stuff together and it's a good thing we are learning that now. On site, it's time to get things started and Team Avatar meet with their kitchen cupboard supplier and their Caesar stone accredited fabricator. Our Caesar stone plans were our plans for Caesar stone. Because we love it so much, we want to go, wow. So Jonathan and Andrew, we went with them, hit them with the Rolls Royce, what we are actually aiming for, partitions in Caesar stone. Can it slide? Can we let it stay still? Can the table go in and out of the island? You know, what can we do with our gadgets? Absolutely. And what can we do with the timeframe? I'm feeling confident. I'm very happy with the choices. There is a few challenges that they've given us in the kitchen. I also like the idea of the blind that they're going to put across the window here as well. I think it might cover the doorway and the fridges as well. I definitely work with the right people that are willing to go the extra mile and put in the extra hours to meet the deadlines. Yeah, it's a tough call, but we're going to do it. Yeah. At four ways, paint and hardware, Team House and Leisure put their colour psychology lesson into action. He's pretending to be looking at colours. That should be it. Hey, wait a minute, I'm pretending. Look at all the colours I came up with. The colour gets to you. What is this? This is the silver bed that we've been using throughout the house. And then we can look at those three options and see which one we can work with. Is this for the wall? Yeah, the back of the kitchen units. Because we need something that will frame the kitchen units and let the grey stand out better. Since you couldn't stop speaking about Amadeus at Dekor X, I thought we could use it as our main colour for the kitchen. Oh yes, that's a beautiful colour. Yeah, we can work with that. It's a lovely colour. Yes. And I think it will be a great contrast with our cabinets and also our Mont Blanc. Definitely. Since we've been shy with colour. And then we're still going to introduce that natural wood. So that's a nice combination. As we criticised the last time for using yellow, I thought maybe we could try a green, since you love it so much. Who said I love green? That is your green. Are you starting to confuse yourself with me now? I've been your head. No, I don't like it. I think we should still go with that. You're not fighting for that green, sorry. The decision is final. And if someone says it's too yellow, you're not going to change your mind again. No, I don't think it's yellow. We're not going to change our mind because you don't have much of it happening in the kitchen. The reason why I love Amadeus is it is because it's a neutral one. And then when you're looking at it from a distance, it looks yellow. But then when you put it among other yellows, it's not yellow anymore. And it's a warm, comforting colour and I believe that it will complement the space quite nicely. Kitchen appliances have arrived and we're still looking for our things. Foraging. Yeah. But we're glad that we got everything that we wanted on time and we're going to try and manage to fit them inside the cupboard. They're delivering our grinding products. We are so excited. They all arrived in one truck, so everybody has to select what they are. Thank you so much, guys. Washing machine. Oh, please send the dishwasher. With the grinding appliances on site, it's time for our design duos to focus on serving up exquisitely functional spaces. Yay, there's so much excitement building as the kitchens come to life. But it's time for us to take a quick break and when we come back, our design duos seek sage advice from their mentors and still ahead, we find out how life has changed for last year's Winner Home grand prize winner. See you then. Our partnership with Winner Home. The best stone is Caesar Stone. It's great to have you back with us on Winner Home on Afternoon Express on this Friday afternoon. Now, designing a kitchen can be a thrilling adventure, but with so much inspiration out there, it can be overwhelming and one can very easily overspend or let a design venture completely off course. Thankfully, our design duos have seasoned experts as their mentors. What's happening here? We are delayed. So we're supposed to get our cupboards in today, but they have not even painted and they're still plastering and rhinolighting the walls. So, okay, how do you look at it now? You've got a fridge. We have a fridge. We have a multifunctional oven flash microwave. So that saves us on time as well. That's nice. That's great. Yes. Especially with this guy that you're building the place for. Yes, exactly. That is their bachelor. He needs everything to be compact and quick. And quick. And we also have a washing machine and a dishwasher. What's the look going to be like? Because that's where I come in. I like to picture the whole story. So the look and feel we want here is concrete and white. We've gone with the rugged concrete caesar stone, which is beautiful. And it has a lovely honed finish. So it's going to be matte. So we want to create a contrast with our cupboards, which are going to be high gloss. And what are the covers? White. White. You've got your way. She wanted a white kitchen from the start. We want to get our accessories right. And Annamarie is giving us tips on how to do that in terms of quantities of accessories and how it should be packaged. What we're hoping to get from Tian is actually approval for our class dependence. We're trying to avoid the bubble effect again because that will make it too predictable that we're going for that again and again. So the last time we spoke, you had roughly around 12K. Where are we on that? How much have you used? What's left? What still needs to be done? We've planned almost, like we've considered everything. And at the moment after taking everything into consideration, we have 2.5 left. But then we forgot about the lights. Not really forgot, but then... We forgot about the lights. What have you bought so far? We did not really forgot about them. We just bought last. Did you buy the important stuff so long? Yes. You haven't wasted your money on props so far? No. Why are you liking me? I know that you bought something. What have you bought? It's a cake stand. How much was the cake stand? $8,000. So you hopped your budget on a cake stand? Actually it was $800. Considering that you spend all your budget, is there nothing that we can make or do? Is there some creative kind of like things that we can turn into something amazing without spending money? He's opposed to do it yourself. You're not a fan of it. You know if you think of old frames that we can like cover up or collage together, or something a little bit, you know, artistic. Is there anything that we can do around that? I think so. You're not a fan. I think so. No, not that I'm not a fan. So you're going to have to make a lamp yourself tonight. You're going to have to think of ways that's in your apartment that you can reuse instead of spending the cash. Because you don't have it. You spend it on your cake stand. All of that cake stand, I know right? But we don't have the cake for it. Exactly, exactly. So the biggest lesson we've taken from Amanda is lists of J, lists of objects that will flatten your space. Without losing your aesthetic.  Here you are. Your boldness. Yeah. Amanda, welcome to the war zone. That is the kitchen. There's a lot going on here. We've got the guys working 9 to 5. Yeah. It's fantastic. And we think you're doing some lighting on it. Right. Maybe some chandeliers. No chandeliers. We've done what chandeliers. So this is where we started off in terms of the inspiration for the kitchen. And this is where you got the bathroom. Absolutely. And we thought an island with like a caesar stone sita that'll seat 6 to 4. Because we wanted to create a dining room in front of the kitchen. You chose the openness of this space. Yes. Well, there isn't much space. So this is a good way of doing it. And then... We also wanted to do like here with the white and the dark wood legs. Yeah. So as you see there, we've got some wood that's gonna go up. Drilled up into these walls. And we've even got the colour. That's Ivory Pillar. We thought let's go for a colour that induces people to eat. It's different. What's the pink red undertone? I'm so impressed with what you've done so far and what you've created. You've definitely hit the nail on the head. I'm just wondering are there any surprises in store that your personality is going to come through? There's always something over the top. And I refer you to the previous rooms with stand on the other than every single one. That is always a surprise. Absolutely. That there has to be. In our sleeve. But there's a lot more to mirror in our minds because we're bringing in a ladder but that ladder's gonna actually be a giraffe. So when you climb it to get to those top parts you're actually climbing a giraffe. But you had to continue with the animal tree. Absolutely. It's too late to let go of the animals. We've thawed it out and we've reeled the South Asian but had a little bit of razzle and dazzle. Not too much. No more chandeliers. Oh man, but just one. I saw a very big one. It's so lovely. If you are keen to see the winter home houses in person then this Sunday is your chance. The Eye of Africa Golf and Residential Estate is hosting a Heritage Day Festival this Sunday, the 24th of September from 10am. There'll be food and craft stalls and plenty to enjoy for the whole family and even our own Clem will be doing a cooking demo. Plus if that wasn't enough, the three winter home units will be open to view. Go to eyeofafrica.co.za for ticket information and come and experience the largest prize on South African television for yourself. And you can also meet the wonderful design duo's. The prize is your choice of one of the three design duo's homes with premier home appliances by Grundig and luxurious finishes by Plascon and Caesar Stone. It's a prize worth more than 3 million rand and it could be yours. So really, if you have not entered yet now is the time simply visit privateproperty.co.de and vote for your favorite design duo by doing so you'll also automatically be entered into the buy weekly prize giveaway. We're also announcing our latest winner of that buy weekly prize giveaway directly after these. Because if it doesn't improve people's lives it isn't Plascon. Plascon. Designed for life. Search for a home on any device any time, anywhere. Private property a home for everyone. It's almost the weekend welcome back to win a home on afternoon express exclusively on SABC3 Now if you visit privateproperty.co.de and vote for your favorite design duo you not only get entered into the winner home grand prize but you also stand a chance of winning our buy weekly giveaway. The winner of our most recent prize is Steven Harbs from Clouif in Durban Congratulations, your vote for team VC has won you a ceiling post-trapedic mattress and base set with almost 15,000 rand. Now because we've started a new design challenge we also have a new giveaway and keeping up with the kitchen theme the latest prize is a grower Eurocube sink mixer professional valued at over 12,000 rand so enter now who knows your dreams of owning your own home could come true as they did for Palessa Moisi who last year became the grand prize winner of Winner Home Season 3. Winner Home not only launches the careers of the winning designers but it also transforms the lives of the lucky viewers that win the grand prize who become owners of amazing properties. We've had three seasons of Winner Home already and that's three lives changed viewers just watching the show entering online and ultimately walking away a multi-million rand designer home so that's what's at stake here it's a massive prize Donino stands up every week and says you're the biggest prize on TV and it certainly is so it's going to be great giving that away again certainly not something that I ever get blasé about that we can be a part of something so big and for a business like Private Property which is all about marketing beautiful homes it makes sense to be partnering with a show that gives one away A fully decorated apartment worth over 3 million rand at the Valdivia State in Cape Town seemed like an impossible dream for the 4th year student Palesa Moisi but the 26 year old from Bloemfontein entered the grand prize competition and won So Palesa it's been a year since you won a home catch us up it's been hectic but nice as well I've been living here for some months now since I got the home and it's been a life changing moment quite a change in student life in the free state what's the plan for the rest of the year the plan is I'm going to move to Bloemfontein and finish my studies and rent this while I'm away then come back home and enjoy my lovely house So let's get serious for a second did winning a home really change your life? It changed my life big time I know and I love Cape Town now I didn't know Cape Town at first and I'm living in this luxurious life knowing places around South Africa that I didn't know before so it really changed my life Was it really easy to win a home? I just entered once I didn't even know you win a house I just entered because I love the kitchen that many of us are doing and then I was just voting for the kitchen and then then I have a home What would you say to some of our viewers who perhaps aren't going to enter because I think it's a difficult process? You know for this competition you don't have to vote to win you have to vote for the best designers then you can win Yes, that's what I did and I won, only once and I won Well, for less has been really good catching up but any last words for my favourite people in the whole world, our viewers I'm going to give you that camera over there Good luck everybody for this fourth season of Winner Home If it can happen to me it can happen to anybody Everybody loves that winning feeling and it's exactly what our design duos are looking for One small victory that they must attain is working within their budget something we all know they're not great at even though they got a 20,000 rand boost so for the revised budgets Team Habitat allocated 70,000 rand for their kitchen while Team VC allocated a similar 68,000 rand which includes their 3,000 rand reward for winning the master bedroom challenge Team House and Leisure however has the smallest budget with only 57,000 rand and they'll need some ingenious solutions to deliver a kitchen that still looks high-end without those exorbitant costs Good luck you two Now someone should probably take a look at those kitchens and see if any ingredients are missing Well, that person is Mwah! At the halfway mark of the previous challenge Team VC faced some big decisions It's happened again this time round with choosing paint colour for the kitchen wall We're going to test two colours Jungle That's the name of the colour game But we didn't test it Did you buy the paint? Yeah, it's there, but we never painted it Did we paint already? With black? Chocolate The agreement was that we're going to try with the green first and then after go with the black because if it's easier to apply a black on top of a green So in that case it's perfect We can live with it We know they can design a master bedroom but can they design a functional kitchen? Team VC, what's up? Hey, how are you? Listen, the progress looks great The kitchen looks glorious Take me on a grand tour Okay, so this is our sink area We're currently standing in our dishwasher Oh, exactly And as we move on this is our cooking area where we have the kitchen product and the colour scheme You guys seem like you're thinking in black and white kind of We were fighting about the black colour and I didn't like the Tribeca but Leso loves it so we're just going to have to go with it I think it's going to look amazing So it's warming up on you It's coming out alright You're not like about the idea that it almost could be practical Because we've gone with grey and white which can be of a dull I think the black just gives it a bit of a pop Lovely, so from a very bright and colourful master bedroom to quite a neutral kitchen with these blacks and whites I mean how are you guys going to draw the whole home together because we're getting to that point now So the aim was to match everything into natural colours so if you look at all our colours they're all kind of like nature you have greens, you have your orange so it's more of using white and tones everything down so with this space connecting to the lounge we're planning to tone it down a little bit So I'm a man, I'm a practical person and I love the kitchen so as much as I dig the aesthetics in this space talk me through some of the practicalities and let's do a tour of those So imagine you're standing next to the pride bowl and you're chopping and then each day you're going to find your bean Ah okay I see so you're sort of scraping things just going to scrape things into the bean So it's quite more practical on the other side in order to get that practicality here we have your grocery cabinet so everything that you can take from the fridge into those kind of things So we're working with our kitchen triangle which is the dynamic space living where you don't need to move too far to get to what you need that's why we added in our prep bowl here so that you can rinse your veggies and everything without going all the way over there and coming back Trendy kitchens are smart kitchens so where do you incorporate the technology here? Well my favourite part of the technology in this kitchen is our power dock which is quite smart because you can put it away so when you don't need to use electricity you can just put it away and it's quite quiet but you just pull it up and you have all the plugs so you can charge your phone charge your laptop while you're in the kitchen doing your thing Well you too good luck let's make it a second challenge win in a row So much pressure I'll see you at completion date Alright cheers So like what do you think of how things are coming together share your thoughts using hashtag win a home on your mobile devices right now Team VC want to prove that they're not a flash in the pan after their first challenge win but team Habitat and team Halston Leisure want to reclaim that crown so after the break we'll check in on their progress stay right where you are I'm just putting my phone away loving reading all the tweets that are coming through hashtag win a home keep them coming welcome back to win a home on afternoon express now we've reached the halfway mark of the kitchen design challenge and both team Habitat and team Halston Leisure hope that they've got the recipe for success to another taste of victory Team Halston Leisure found themselves in every interior designer's nightmare where their caviar taste didn't match their tinted tuna budget but the bonus cash and generous sponsorship might be their kitchen saving grace well it seems like there's no one here from team Halston Leisure I'm here there you are how's it going nice to see you so what are you doing behind the fridges I was checking out the positioning of the things if they're going to get it right but then otherwise everything is looking good it does look good I like your cabinetry this island in the middle here it looks a bit skew though so how's it going to get halfway mark they're not yet done with the island so it'll be fine once they're done a good designer knows how to incorporate both functionality and good design aesthetic and do that all within a budget so how have you managed to do that in your kitchen functionality wise we decided to keep like everything on the one side and have like our preparation area in the center of the kitchen so in that corner we'll be having our double fridges and then from there you move onto your stove and in that corner it'll be our double sink so next to the sink we have our dishwasher and our washing machine and what about the design aesthetics in the kitchen from a design perspective we chose like the color the plus con color of the air which is the our medias and then we try to contrast that with the dark storm gray wood finish and have a cherry which that material we used in our bedroom and the main bedroom so that ties all those rooms together nice ideas but I'm not hearing challenge winning ideas from you guys in this space what do you believe are the show stoppers for you because everybody's got an island everyone's got paint on the wall and everyone's got cabinetry what makes yours so unique I think what makes it unique well I haven't seen the other guys' kitchen but then simplicity is always key that's what I believe from a design perspective not complicating the life of the person who's going to use I like that all right so what's the needs to be done before we get to completion what needs to be done they don't need to paint us getting touch up the ceiling line and our hope still needs to go in scissors stone tops needs to go in there's still quite a bit that needs to be done our doors are not in yet so I'm trying not to panic because finally it's not here I'm buying myself today so I need to keep myself together well don't stress too much get back to the work I want to see a simple kitchen by the end of this all right we are in good stuff good luck dude thanks team habitat are hoping to create a kitchen that is the toast of the competition hello team habitat welcome to the kitchen watch cooking in the kitchen sure well unfortunately we could make you some Sunday with some wood cream but we're saving the wood cream for our wall is that the name of the paint that's the name of the paint but you guys can serve me up some drill bits and some screws and some wood that's all we got right now some shelving cabinetry metal things measuring up here well listen it's only the halfway mark you guys got lots of time to get this complete but it's looking like you're making good progress what's the idea in your kitchen so we wanted to create an open kitchen because the bulkhead dining room is going to be a big feature so we're going to bring in Caesar stone from where we're standing all the way to the end of the bulkhead and then have some seating on other side so basically we've just lowered the ceiling to create a focal point for the kitchen and a dining room extension so it's basically just highlighting the main entertainment area and it's going to be highlighted once that lightens up not even a neon highlighter could compete with this bulkhead and our lighting however this bulkhead extends all the way from the kitchen quite deep into your lounge area which is another space for another challenge we wanted to use the open area space the kitchen and lounge to not just have two rooms but three rooms because you have a kitchen, you have a dining room and then you have a lounge which is all open play out buy two and get the third one free I don't quite know if I've bought into that but it's a very small space and you guys are cluttering with lots more extra bits and bobs and a birdie has also told me that you guys have got other little surprises to add to the space like you must have the birds to start chipping but we do have a few surprises like we've got a ladder coming in to help you get up and down to the putts right at the top because we wouldn't want you to jump up and down like a netball player it's not an aerobic this is a functional kitchen a giraffe come in yes you're going to carry with you the animal theme it's only right if you've got a zebra and a lion's den there needs to be the tallest animal in the land but the question everyone wants to know from you two what's happening with the lighting we're still in debates but we're getting lighting definitely to light up this bar cave because that's the point of having a beautiful bar cave if no one can see it and it seems that the word chandelier is for better so let's say we're going to light it up with some chandelier lighting and we're going to buy it at the lights one thing that was not raining in this competition was obviously the money Brad has he been looking after your budget? we like my waistline I've been training in a fine land we've been watching that budget but it's been a bumpy road I'm typically saying we've behaved and we've stayed within the budget and tried to make everything fall within that so that's how the minimalistic feel of the kitchen has come about stunning deadline is approaching fast what still needs to be done sure the caesar stone needs to come in between the caesar stone tops the caesar stone dining room obviously we don't have doors as you can see on our cabinets that's not part of the plan the doors are definitely coming in with door handles back splashes are coming lots still to be done I was always limited I do not want to see you stressed at the completion date good luck you two and I expect a milkshake at that completion absolutely with whipped cream ciao show let's get back to work well for now all we're looking at are some half-baked kitchens but next week we hope for some steaming hot completed designs of course our judges will be joining us our country will be all sorts of spicy but the big question is which design duo's kitchen will be the cherry on top you'll have to stay tuned next time to find out we'll see you next week Friday from me Danny Laquissa have a great weekend bye bye search for a home on any device anytime anywhere private property a home for everyone with love by clover never feel good production


Okay, cool. Okay. Let's get started show you. So this is the project update for the documentation team We look after the open stack manuals our post tree. So my name is Alexander settle or a settle on ISE I am the PTO for the pike release and these are my colleagues. If you'd like to introduce yourselves Yes, I'm all gone. I'm a call reviewer in their role One-stack the commutation upstream. So I'm with their community with open stack for three years already It's great time. Thank you So we are here just to present the update of what we plan for the pike and for We have the third so the picker it's my teammate, okay Thank you, Olga. I'm Alex. I'm happy to see you here I will talk a few words about writing security documentation Impact and the importance of this activity for the project for the upcoming release project So this is the three of us. There's a team of probably about I think it's probably about ten or twelve cause We are quite a big core group, but we're not all full-time I work like of all full-time working upstream. I should say So I've been working on the project for about three and a half years now. So since the ice house release So just been a little while now. Anyway, so As if you've been to these project updates before you will know that we're gonna go through a little bit of what's happening in pike Which is the current release? What we're planning to do for the following releases, which is going to be Queens and Rocky And then we're just gonna have a little bit of questions and feedback So we're gonna kind of keep this is like not the most in like formal thing ever I know that it's recorded, but you know, it's just kind of make it chill. Let you know what's going on and move from that So what does our documentation team do so essentially like we are officially the open-stack manuals team And we currently look after the open-stack manuals. These are all of our repositories So we do the API ref security doc training guides and training labs The API ref security trade and the training sections are actually all under different specialty leads So if you sort of look at it like an umbrella, there's me at the top We have our specialty leads and all of our cores which look after the Ups-deck manuals So we actually have quite an extensive amount of documentation to look after we have user guides admin guides Networking installations Deployments, it's quite a vast selection of stuff We will continue So what do we do? We pretty much focus on the creation maintenance and organization around docs.opens-deck.org So if you've ever been to that site before it is Quite large, but we this is where we keep all of our main documentation and it's actually one of their most searched sites for open-stack So it's open step wwb.opens-deck.org first and then it's docs.opens-deck.org So we have a really large customer base, which is really important for our users and operators Okay, so we were unofficially founded in the Austin release And then we were officially from Bexar. So 2010 ish It was actually started by Ann Gentel who is still a major contributor and she actually looks after our API references We had 162 contributors for the latest release As you can see here, the 61% of the community members use documentation daily or weekly. This is really big It means we have a lot of people coming and looking at what we're doing Which means we're in the spotlight all the time. There is a lot of pressure to update things continuously But we have a solid team looking after that and we have a lot of the community is really supportive We have Liaisons so we have doc liaisons. So like the Glance team have a liaison for documentation for us And we try and make a community the community all banned together to make it happen so Mike, Olga Okay, so it's quite obvious that the major focus of All the documentation is user experience course docs is user experience Also, we Some work on my manageability as well Now I'm going to move on With there are a new features and enhancements that we plan For the pike. So the first one is Achieving all the documentation on Docs or OpenStack orc this So this addresses the requirement to achieve all the docs that That is not I'd make it Available to the users who are who need it course. We know that Lot of people just come to us and say that today need the other eyes also at least and but Yes, it's the problem So what is the plan we have we have the Specification of red the ready already the plan is to That at every new release there at least that it comes and of life We just make the The folder that is read or read only that that can be downloaded by the user Built a law locally as well as make it available From the from the docks or post at org another Enhancements is moving the administrator guide to project reapers as you may know that the number of projects of OpenStack projects is getting Begin and bigger with this release but the docs team is shrinking just so we need to keep keep them up up to date and to Deal with the problem we decided to move them to the project Poder repositories just the same way we did it with the installation guide and References the sub as the sub tasks for this enhancement for this improvement is also set and up a cookie cutter For all we know from the in the install guide and deployment guide tool for developers as well As so check in cleaning up and that is in the content of the guide to to align the There are content with their conventions that we have in their Contributor guide for documentation Um, I'm on then other new features is their continuum revision of their Architecture design guide just to make the other content Just to to create the obstruction between the cloud Architecture core concepts and various OpenStack pro objects So and the last Probably not the least in this release and enhancements is adding The support for for Reno That was one of their major tasks From Atlanta PDG Thanks to Adam Spears Him him made it of it Very available him he made it happen and now our repository Support Reno so I Also to I want we decided that it's better once once to see Then here for several times and I want to present just as a small demo to To show how it works. So there are you you use cases very simple It's it in the result. We we will recreate their It is no note reports report under the under the subsection of the Release notes. So let's let's start our first you need to install the rena tool Just to have it and On on your machine, and then you install the sphinx in x extension that is built in it And and rena you would see it to build Their notes Once you've done it You need to Took took long their repository. It's all Let's take my nose in our case. So you change the directory to the repository and Here I will need to their it is not that directory.  Listeners reports report. It will create a yaml file And that will be stored under the list now knows that directory actually That's successful So you check here read the file that we created You can see it So now we are need to add their content our tool this file so you open the file and Edit the template that is already there with all of the sections. I decided just to add some text some tool tech on tributary guide sec section so We edit it with we save the file and Now it's very important before you built To see the output for your use to talks you need to add to Commit your your change course talks will will Use the locks to build the other output So you you can unmute your your change Create the Garrett Yeah, so It there are you Now I will recommend that you check there the output you was in talks comment You can view it locally In the build to build directory that is Recreated automatically once you run talks so open Unreleased and in there Exciting things about what we've done with Reno is usually you'd come with a set of a Format a template that was just simply like updates config We worked with the Reno team and specifically Doug Hellman and Adam Spears to actually make this Completely separate for docs. We made sure that we're actually able to update per book rather than per project So that was just I'm sorry. I just wanted to quickly add that Change I checked that the time it ends in place marked it now Okay, I'm I'm market as a broken progress not to to confuse the reviewers and in a while you just can check The build So it's in place. It looks good Under there as I remember we created the one under the Release subsection. So it's here Okay, back to the template. Yes, it's good and I Think that we have all the ear ears in place here from the Changers that are we are visible to you through the users so that that can be you some changes in you are and on The site as well as some some in in internal chair and just like translations or something like that and also the template includes the section that that are indicated to The guy through the guides something like user admin and and and so on so forth So and if you want to add some some section, you just need to To commit the change not to the project itself here But but for for this template to go to code on figure. Yeah, I'm a file. So that's the workflow. Thank you So That's that one previously We were actually building all of our release notes manually which meant at the end of each release It was up to the a specialty team leads to build it one of the problems that we've been having is We've been losing like major core contributors to the documentation project Which has led to less people being able to form these specialty teams Which then has also meant that we're losing, you know, we're losing the ability to like have concentrated efforts So we're trying to re-centralize and having the Reno support release notes was really important for that I'm having something automated that our contributors could directly can contribute to So I just also wanted to talk about this one last thing that we've actually worked on at a very last minute This is a governance documentation tag. So governance has tags to govern projects We actually previously haven't had anything like this for documentation It's this is just designed that Okay, let me start this again because I'm rambling in circles Basically, what happened was is it got to a point where our installation guides? We're not being verified by anyone outside of the documentation team, which in fairness is our domain But it is becoming less and less our domain because we've been pushing the installation guides back on the project team And we're about to do the exact same thing for the admin guides this is important because a Doc information per information developer or a technical writer has a base level of information on every project They do not have the in-depth knowledge that a developer and a key operator and user has on that totally different side So we can easily go through and verify these guides by just going click step by step But a full manual install takes several days And what was happening is we were finding a lot of failures later on because we were there was a communication There was like a distance in communication between what was happening in the projects versus what was actually being documented upstream And because essentially we are the official documentation that was a big problem when no one can install and no one can start up an instance so I Through a little tantrum and I went up to the government's depart Governments repo and I was like we need to make this official We need to have a check that these projects are actually making sure that they are verifying this and what this means is that They get this tag associated with their project once they have verified their guides So this all applies to not only the installation guides But the admin guides that we're also going to push out as Olga was mentioning So this is going to be really important for like the future This is specifically for our users and also like our operators and developers as well Just so that they know that this is going to be important to our future But this was a bit of a random last-minute thing after we've kind of had a couple of releases now Where things haven't quite gone as smoothly as they probably should have So we're trying to govern it a bit more and make it a bit more Easier basically so Alex is going to go through Queens now Unexpectedly Queens has Continuation of the activities started in pike But what's what's different is you can see the major focus is on security during this release as you know open-stack documentation has Security guide created With the help of Rob Clark's security team Unfortunately, it's been created like for for many years ago and need to be redesigned reconstructed taking situation new features and new threads we have and That's why We have linked to either part where we collected the main problems Found in my security guide and the new table of contents so you can actually take the participation In this in this activity as well Um, luckily, I was involved in writing security best practices for merantis and this sounds pretty pretty much similar and I can emphasize that actually bad things happen by day two. So when it's everything is deployed it works well, but Some vulnerability it can be found and Administrator need to think how to how to fix this one with how to patch apply the patch and Here our security guide comes Yeah, thanks it's obviously to release time from now and This is a little bit harder to think this far forward because one of our biggest things is user experience Documentation is all about the user who's reading it. Who's our customer and our customer is always the operator In saying this, you know, our biggest problem right now is what is our future? So we've been you know today has been really big for us at 12 o'clock We had a session on you know, what's the future of the documentation team? How are we going to keep continuing maintaining our docs the way we currently are? Historically speaking, we've always had a team that has been able to dedicate time to this process and we've also had plenty of one-off contributors or Irregular contributors from different projects that are able to put in the time and effort to help us But this is becoming less and less Available so we've had a lot of discussions about where we're going to go We're trying to automate as many documents as we can so we're already looking at automating our configuration reference and our command line documents We're looking at creating new scripts. We're also looking at pushing the administration guides and all of the Install guide back to the project repose and with that we are then going to switch the process of most likely This will be a specification. So I'm definitely giving you insider info as of from 12 We're going to look at instead of there being a liaison from the development team to us We're going to be liaison specifically to the team so as an example, I would look after the Nova team, but I could also then do three what are called optional projects so that way I could maybe look after designate Swift and The banana project That doesn't exist So, you know what? I mean like we're trying to kind of come up with new ways of doing this thinking new even things But this is our key focus for Rocky I'm not going to stand here and tell you that we're going to have all these amazing things happening because right now Our main focus is what's going to happen? We have an amazing documentation suite and it can all be maintained by the like the Community, but we need to be looking a little further ahead now We did start everything in Austin and technically Bexar, which has been a really long time But that doesn't mean It's going to keep going full steam ahead the way it is We have to start thinking about the future and how this is actually gonna like in how we're gonna actually make this work Because I mean the amount of times Had a lot of people say, you know documentation is so important documentation is so important But you that doesn't really come through at the end of the day, you know People are really happy to say what they want to say to your face When it comes down to it, you know, we really struggle to get contributors and realize that documentation is important I could stand up here all day and tell you about how the users need documentation and without the users You do not have a development project. I mean you do but no use it But you know that aside, this is our focus So yeah, that's kind of where we're going and we'd love to have people along for the ride and working on it with us We've got some cool things coming up. We're archiving. We're moving back But we're still also keeping our core team together and making sure that we're a centralized team So that's really important to us right now But yeah, this is a slide that they kind of made us do I'm sure you guys have kind of seen it by now, but it's pretty much just like a couple of questions I mean if anyone Can help us anyone who operates and uses open stack and has used our guides before love to hear from you But if you haven't that's okay as well, or if you just have other questions that works as well But this is kind of where we're at right now. What we're thinking and this is just Like random questions. This isn't you know any targeted anything. We're just trying to figure things out where we should be going You got any questions or you got any bright ideas? I'm right here Thank you. I Wonder if opensack.org has some analytics that will help to figure out like how our users They reach the documentation whether they enter the main page main documentation page doc doc's open stack.org. Yeah The other is like special separate like flow users for flow that will help to figure out how how how users get the documentation What what what they're looking for actually? Spend a lot of time looking down to is that our install guide is absolute top So when if these analytics basically show that the as far as open-stack concern the top hit goes to open-stack.org And the second top hit is docs at open-stack.org and that is by a million views It's it's really intense, but that's also great. It means that you know people are viewing it and they want it And they're interested And then from there we've noticed that it's our install guides and our networking guides They are absolute top, but networking is it's very specialized We don't have a lot of people that are able to do this intense networking configurations Because it's not just down to the neutron team. It is so much more than that and it's really quite difficult So I mean we have all these numbers and we're able to point You know the Chinese contingent is huge our translation team Works, you know over time to get the Chinese translations out. Whoopsie daisies. Sorry guys We've most recently moved to using PDFs as well Simply because the Chinese contingent definitely wanted this this was important for them They wanted to be able to download it and hold on to that document rather than having to access the internet constantly So yeah, I mean This that's a that's a good and they'll point us in the right direction sort of But it can be a bit. I don't know if yeah, I mean, it's funny. We can look out here I've got them all downloaded. I've got a lot to go through But yeah Yeah, can I click on some go for it, please if it's not secret? No, I think that is real time. Okay Yeah, any thoughts any questions or are you guys all just here because it's an empty room if you are that's okay Thanks, Amy Yeah, anyway, I don't have anything else. I don't think unless anyone has any questions about Project stuff, but if anyone's interested we have an operations and administrations guide moderated session tomorrow we're going to be looking at the overlap between these guides because well administration is for operators and the operations guys also for operators But one is slightly more theoretical and one is definitely more practical But it is definitely maintaining two things at once that probably don't need to be so we're kind of trying to hopefully reach out To our operators and see what they actually want rather than guessing So if anyone's able and interested to attend that that would be great We also have a session on developer.opensduct.org Which is our API site which we look after all the API and SDKs as well So if anyone wants to come talk about that as well, we'll be doing that on Wednesday We'll be looking at you know, who's our audience? What do what do they actually want because at the moment nobody seems to know Which seems to be a common problem with OpenStack Yeah, cool Excellent love the blank faces Thank you


OK, so we can start. So we were talking about the CELO theorems. So we briefly call the statement. We had a P-CELO group. So if G is a group of order, say, P to the m times R, where P and R are relatively prime, a P-CELO subgroup is a subgroup H in G, whose order is the maximal possible power of P to the m. And so the first CELO theorem says that such subgroups always exist. So this is the first CELO theorem, which says G has a P-CELO subgroup. OK. And so I will still, because we're using it a few times today, recall the corollary, the main first corollary, which is Cauchy's theorem, which says that if a prime P divides the order of the group, then the group contains an element of order P. So then we had made a few first applications. And now I wanted to come to the following, which I had just stated but not proved the last time. So this is the following theorem. So every group, actually, I maybe have to be slightly careful about the statement the last time, so let P be a prime, which is at least three. And then every group is looking of order to P is cyclic or diheter. So it means it's either equal to the cyclic group with two P elements, or it's the diheter group Dp. And so I just, again, recall the statement the diheter group Dn is a group of order 2n with the following two condition, which is generated by elements Ap, such that A to the n is equal to 1, B to the b squared is equal to 1, and A, so Bab or Bab to the minus 1, whatever you want, is equal to A to the minus 1. And you can easily see that these conditions allow you to write down all the elements in the group. And so it's determined uniquely by this up to isomorphism. So let's try to prove this. So we will meet the use Cauchy's theorem. So this thing has two P elements. So in particular, P is a divisor. So let G be our group of order 2P. So by Cauchy's theorem, there is an element of order P. So G contains an element, which I call A, of order P. And also B is the divisor, so also 2 is the divisor of 2P, obviously, and an element B of order 2. So we look at the subgroup generated by this element of order P. So this has P elements. So as our group has 2P elements, we have that H is a normal subgroup. No, H has index 2 in G. And it is a subgroup. So by an exercise, we find that it is a normal subgroup. So we have found this wonderful normal subgroup. So therefore, if we write BAB, this is because B squared is equal to 1. This is the same as BAB to the minus 1. So this is the conjugation of A by some element in G, actually by B. So it follows that this must also lie in H. So also this thing must lie in H, because H is a normal subgroup. So this is equal to A to the I for some I, because it lies in the subgroup H, which is generated by A. So let's see what we can say about this I. So we can, for instance, do it twice. So we take B squared A B squared. This means we apply B. So obviously B squared is equal. So A is obviously equal to B squared A B squared, because B squared is equal to 1. On the other hand, we can also view this as B, BAB, B. So this is equal to B, A to the I, B. And you can easily see as B squared is equal to 0, we can write this also equal as BAB to the I. You just write this I times next to each other, always between two factors, you have B squared, which is equal to 1. And so this is the same. So this is A to the I to the I, this is A to the I squared. What? What? A to the I. Yes. So if you write this, BAB times BAB. This is BAB, BAB, and this is equal to 1. And so you just know. So in other words, we have, so it follows, that A to the I squared minus 1, so if you multiply it I squared minus 1 with itself, this is equal to 1. We just divide by A on both sides. Now we know H is a cyclic group of order P generated by A. So we know that A to the, A to some number is equal to 1, if and only if P divides that number.  And as P is a prime number, if it divides the product of two integers, it has to divide one of these two integers. So it follows, I may be right now, divides like this. P divides I plus 1, or P divides I minus 1. Now I want to look at these two cases. We will find that in one case, the group is cyclic. In the other one, it's the hedger. So let's see. So first look, so if P divides I minus 1, what do we have? We have, and it means that A to the I minus 1 is equal to 1. So it means, if we multiply by A, we get A to the I is equal to A. But that just means, A to the I was BAB. So BAB, which is the same as BAB to the minus 1, is equal to A. So that means BA is equal to AB. So A and B commute. And as our group is generated, so let me see. So A and B commute. Anyway, so let's say not go too fast.  So yeah, let me. So if A and B commute, you can say that obviously B is not in the subgroup generated by A. So it means that A and B generate G. Because if you take, so this contains already P elements. If you also add to it the element B and all the products with it, you get two P elements. So G is commutative. Because the two elements will generate what? I didn't get the word. You have to say speak louder. I don't know what you said. So I mean, maybe it was a bit fast. But so we have H is a subgroup which has P elements. B does not lie in the subgroup. And so therefore, B does not lie in H. Because B has order 2 and this has order P. And so it doesn't lie in that. So therefore if we, so the subgroup generated by A and B must have order at least 2 times P. And so we find that it's generated by them. And these two elements commute as we have seen. So G is commutative. Because you just write any word in A and B and then you can they commute. So we have a commutative. So it means in particular that so thus we also have the group subgenerated by B is a normal subgroup. And we have that intersection of these two subgroups consists just of the unit element. Because all elements different from the unit element here have order P. And all elements different from the unit element here have order 2. And this cannot be the case for one element. And also the group is generated. And so we can, and by the argument I just gave, we have that if we take all products of elements in A and B, this gives the whole group. So we can apply this theorem we had before, which tells us that, so by previous theorem, which tells us under what conditions you get a product, you have that this is the product of these two groups. We have G is equal to A times B. But this is isomorphic to a cyclic group of order P. And this is isomorphic to a cyclic order group of order 2. And then if we want, we can apply a previous theorem, which tells us if the two such numbers are relatively prime, in this case P and 2, then the product of the cyclic groups is equal to the cyclic group of the product order. This is isomorphic to Z, module 2P. So we find in this case it's a cyclic group. And here we have used this case. We have the other case that P divides I plus 1. So well, we just look what we have. So that means what? So then that just means that A to the I plus 1 is equal to 1. So in other words, if I divide by A, A to the I is equal to A to the minus 1. So thus we find that BAB is equal to A to the minus 1. We already know that A to the P is equal to 1, that B squared is equal to 1. And we have assumed that the group has 2P elements. And again, by the same argument as before, the group is generated by these two. So we see that this is precisely the condition that characterizes the Tahitian group. So the group is the Tahitian group into P elements. So it's Dp. So it says the Tahitian group Dp has 2P elements. OK, so yeah. Well, in some sense we can, I think we can already say it from the beginning. Because we have a group of order 2P. We have an element regardless of the commutativity or anything like that. So we have that A is a subgroup of order P. And we have B an element in the group, which is not in the subgroup. So it means that kind of the other coset with respect to this is so it really implies, so we have A as P elements. And we have the group, and we have B is not in A. So we could make, we can just look at the cosets for A. So we have that G is equal to A plus B, so union B, A. Now these are the two cosets for the set. So it means that actually every element can be written as an element as a power of A plus B times a power of A. So in particular, the group is generated by A and B. This is independent of commutativity or anything. It's just by the fact that this thing has indexed 2 in the group, and we have found one element which does not lie in the group. So I didn't maybe explain this so properly, but it's a very simple fact, doesn't have anything to do with the rest. Is it clear? Yeah, that's true. So that's what I, why do we know it's a cyclic group? Yeah, yeah, no, that's what I deduced. But you say you can see it more directly. Yeah, okay, yeah, maybe you're right. But I don't quite see. Yeah, yeah. So if you just take, so you just say if you take the powers of A, B, you can get all these elements. Yeah, okay, no, that is in some sense true. If you just look at all the products and you see what you get, you find all the elements. That is true, yeah, okay. Okay, so I needn't make such a complicated story, but okay. Okay, so now we want to come to the second C-lo theorem, which is a bit more complicated to state, but so it says maybe something about how the different P-groups and how the different P-C-lo subgroups are related to each other in some way by conjugation. And so the second C-lo theorem says in particular, so one kind of consequence of it, which is easier to remember, that all C-lo subgroups are conjugated to each other. But the precise formulation is it says somewhat more, which makes it also a bit more complicated. So this is the second C-lo theorem. So what does it say? Let's say if we take any subgroup, so yeah, of G, whose order is divisible by P. So let H be a P-C-lo subgroup of G, then there is a group which is conjugate to H such that the intersection of it with K is a P-C-lo subgroup of K. So then there is a conjugate subgroup H prime equal to say G H G to the minus 1 of G such that K intersected H prime is a P-C-lo subgroup of K. So in particular it somehow says how you can get P-C-lo subgroups of subgroups by just taking a P- C-lo subgroup of the big group, conjugating it and taking the intersection. So we want to give a couple of corollaries. So first, let K G be a subgroup, which is a P-group. So that means the number of elements is power of P. Then K is contained in a P-C-lo subgroup of G. So largely, obviously. So you will always be contained in a group with the maximum P power of elements. And the second one is the one I said before. So the P-C-lo subgroups of G are all conjugate. I mean, you should notice that if I have a P-C-lo subgroup, if I take a conjugate of it, it will also be a subgroup. And as a conjugation is a bijection, it also has P to the M element. So it's also P-C-lo subgroup. So all the conjugates of P-C-lo subgroups are P-C-lo subgroups. And here we have the converse that the P-C-lo subgroup. You can also get all the P-C-lo subgroups as conjugates of just one of them. What does maximum mean? It's not contained in. So it's the biggest possible P-group. But there can be bigger subgroups, I think. But that's kind of trivial anyway. Because this P-C-lo group has the P power of elements. So there can be no other P-group which is bigger. Because the maximal number which divides the number of elements in the group is P to the M. So the maximality follows just from the number of elements. But it is true. It's also maximal. But it's a little bit like a statement that everything is contained in. It's maximal in the stronger sense that everything is contained in one of them. So that's kind of OK. So now let's first prove this corollary. So I just write also down that as a conjugation from H to G H, G to the minus 1 is a bijection, we have that a conjugate of a P-C-lo subgroup is a P-C-lo subgroup. So that I said before. So let's start with proof of 1. So if K is a P-subgroup of G, so it's a P-group, that means the number of elements is the power of P, a subgroup of key, and it's a subgroup of G.  It contains the maximum P-power of elements. That's kind of a P-group. It's a P-C-lo subgroup of itself, kind of tautologically. So we can apply the theorem to this wonderful situation. So thus by the theorem, so the second zero theorem, if H is a C-lo subgroup of G, then there is this conjugate and so on. So then there is a conjugate H prime equal to G H G to the minus 1, such that if I take the intersection K intersected H prime, then this is a P-C-lo subgroup of K, actually the unique P-C-lo subgroup of K, because obviously the only P- subgroup of K, the only P-C-lo subgroup of K itself is K itself, because test of all the elements. So I could also actually be unique. So this must therefore be equal to K, which is nothing else that K is a subgroup of H prime. And as H prime is a C-lo subgroup, we find that K is contained in it, which is, so H prime is a P-C-lo subgroup. And so we find, kind of by this trivial application, that this first one is true. Now let's look at the second. So we were right here containing K. So let's look at the second. So what is the statement? The P-C-lo subgroups are all conjugates. So let's take two P-C-lo subgroups of G. So then there exists a conjugate, say H prime equal to GH, G to the minus 1 of H, which contains K. So this is by part 1. K is a P-group. And we have said that every P-group is contained in, actually by the proof of part 1. In the proof of part 1, we have just taken, we've seen that not only it is contained in a P-C-lo subgroup, but for any given P-C-lo subgroup, it's contained in a conjugate of that, and it's proof. So H contains K, but H has P to the m elements, and K has P to the m elements, so they're equal. So H prime contains K. H prime contains K, and the number of elements in H prime is equal to the number of elements in K, which is P to the m. So they're equal. So we find that, indeed, all P-C-lo subgroups are conjugate. So this is a slightly complicated statement, but that's how it is. And maybe the corollary, which just says some special cases of it, is slightly easier to understand. I mean, keep in mind. I want to directly go to the third C-lo theorem and then give another application. So the first C-lo theorem says that there are always P-C-lo subgroups. The second one says in particular that they are all conjugate, and the third one says something about how many P-C-lo subgroups there are. Third C-lo theorem tells you not quite completely how many there are, but at least gives you some idea. It tells you something about how many P-C-lo subgroups there are. So the statement is the following. So again, I repeat the usual assumptions that we have that G was a finite group of order P to the m times r, where p does not divide r. Yes? So let S be the number of P-C-lo subgroups. And we can say something about this number. We can say two things. Namely, then S, this number of P-C-lo subgroup, divides the number r, so the rest here. And S is congruent 1 modulo p. So it's S can be. So it has these two conditions. It's a divisor of r, and it has the property that if you divide, take the rest of division by p, you get 1. So this is a rather strong restriction. So in particular, if the numbers P and r are somehow not too large, very often this will be almost impossible to fulfill, or it will force S to be 1 or something. So let's give an application of this where one actually precisely uses that. And this is another characterization of many classification of many groups. So the statement is, let G be a group of order, so G, which is equal to P times Q. Where P and Q are primes, P is bigger than Q, and P and Q are primes. So we have two prime numbers, P and Q, one is bigger than the other, and in addition, we have, so if Q does not divide the number P minus 1, then G is cyclic. So if we have this kind of, so for quite a number of possible number of elements of the group G, we can completely say what the group is. Up to isomorphism. So this is true, for instance, if the order of group is equal to 15, 33, 35, and hope 35 is correct, and so on. So there are many cases. OK, so let's try to prove this. And this actually is an application mostly of the third zero theorem. So we take NP to be the number of P zero subgroups. So we have these two primes, NP equal to the number of P zero subgroups of G. We want to find out what this is. We know that by the third zero theorem, we have that NP is congruent to one modulo P. And so yeah, OK, I wrote the number symbol, but number of, I mean, sometimes one just had this to be number, but maybe I shouldn't, at least not without warning you. So NP is congruent one mod P, and it's supposed to be a divisor of Q, but P is bigger than Q. So if it's congruent to one mod P and a divisor of Q, it must be one. Because if it would be already P plus one, it cannot be a divisor of Q, because it would be bigger than Q. And in P divides, I write it now like this, Q. So as P is bigger than Q, it follows that NP is equal to one. So there's only one P zero subgroup. But we know that all P zero subgroups are conjugate to each other.  So it follows. So if G is an element in G, then G H G to the minus one is it so? So let H be the unique P zero subgroup. Then if G is in G, then we have G H G to the minus one is also P zero subgroup, but there's only one. So this must be equal to H. So it means that H is a normal subgroup. So we see that if there's only one P zero subgroup, it always has to be a normal subgroup. So this is actually we have done by applying the second. Well, not really. And now let's look at the number N Q to the number of Q zero subgroups. We will want to show that this is also, this number is also one by a similar argument. So we know that N Q is equal, is congruent to one mod Q. And is it correct? Yes? And P divides the number. No, and N Q divides P. But now, so N Q has to divide P. So P has only two divisors, namely one and P. So it follows is equal to one, or N Q is equal to P. Only two possibilities. So we want to see that N Q is equal to one, so we want to see that this is not possible. So if N Q is equal to P, we know that N Q is congruent to one mod P. So then P is congruent to one mod P, mod Q, obviously. Sorry for the misprints. And so it means that Q divides P minus one, which is precisely what we have excluded here. We have excluded precisely because we didn't want this to happen. So thus it follows that N Q is equal to one. So this is a contradiction. And so N Q is equal to one. And so now the number of Q zero subgroups is also equal to one. By the argument we have made before, it means that the unique Q zero subgroup is a normal subgroup. So K of G is a normal subgroup. So note that these are H is isomorphic to Z mod PZ, because it is a group with P elements and every group with P elements is cyclic. And K is isomorphic to Z mod QZ. In particular, every element in this thing, which is different from one, has order P. Every element in this one, which is different from one, has order Q. So we find that H intersected K consists only of the one element. And so we can apply a previous theorem, although maybe we have to explain a little bit. We had this theorem which said that if we have a group and we have two normal subgroups whose intersection is one, and if you take the product of any of them, it's the whole of the group, you get that the group is the product of these two normal subgroups. But if you look at the proof, we had shown that the map from the product to the group is injective, is a homomorphism, it is injective. And we used the fact and surjective. So for the injectivity, we used only this. And for the fact that it was well-defined, only for the surjectivity, we used that every element in the group is a product of one element of one of the other. But here, we have that the number of elements in the product of the group is P times Q. So we have an injective map from H times K to our, an injective homomorphism from H times K to our group G, which must be bijection because they have the same number of elements. So this means they are isomorphic. So I should maybe have stated that when I did the previous theorem, as you also kind of hinted in some sense, but anyway. So it follows by previous theorem and its proof that G is equal to H times K. So if you have this thing, if you have two normal subgroups of a group with this property, then G is the product. So that means that is equal to Zp Z mod P times Z mod Q, which as P and Q are two different primes, is isomorphic to Z mod PQ PQ Z. So this is the last application I want to make. So what you can see is that at least if the group is not too big, this fact, these zero theorems can actually tell you quite a lot about the group so that in many cases you can actually find out precisely what the group is. But one has to be slightly, if you look at it, in all our cases we somehow always had assumed that the number of elements in the group has very few prime factors. And it kind of gets not so nice anymore. And you cannot say so much or so easily when you have, for instance, some high power of a prime. For instance, we do not know what the groups of order 8 are. It's not so difficult, but from what we have said, we don't know it, or 16, or 12, whatever. OK. So now, obviously, I will have to prove these theorems, the zero theorems. So I've made all these applications. So somehow, you might want to be sure that the theorems are actually true. So let's start with the proof of the first zero theorem. So we start with the elementary lemma, which is not really about group theory, but just about integers, about counting things. So the number of subsets of order p to the m, so with p to the m elements, of a set with, say, n equal to p to the m times r elements, where p does not divide. r is given as follows. This has nothing to, actually, this is independent of this assumption, so n choose p to the m, which is n times n minus 1 until n times p to the m plus 1 divided by p to the m times p to the m minus 1, minus 1, and so on. So the typical element will be p to the k minus, p to the m minus k, for some k and so on, until we find the n with 1. So everybody knows these binomial coefficients, and you know, I hope that they can be defined as the number of subsets, so n choose m is equal to the number of subsets of m elements of a set of n elements, or the number of subsets of m elements of the numbers from 1 to n. So this is more or less how it comes up, if you look at the binomial formula, it comes precisely up by counting these. So this is standard. So this is, I'm not going to prove, because I consider it as standard, but the new thing is that furthermore, p does not divide m, this number n. So the binomial coefficient is not divisible by p. OK, so let's prove this. As I said, the first one is elementary, and I will not end well known. So that this number of subsets is this binomial coefficient is well known. And sometimes one does it in school. So we have to show only this last thing, that p does not divide n. He does not divide n. So here we just have to look at these factors. What I'm claiming is that whenever some power of p divides a factor here, it divides with the same power the corresponding factor here. So for instance, here, obviously p to the m divides this one, but p to the m also divides this, and it's the highest power of p which divides it. So there's no power of p which occurs here. And you can see that it goes on like this. So let me write it. So the claim is whenever, or maybe this is the note, whenever p to the l divides n minus k, then also p to the l divides p to the m minus k, and vice versa. This is just because n is equal to p to the m times r. So the same power of p which divides this, if you take the highest power of p which divides n or n minus anything, namely because this is if and only if, because this is the maximum power of p that divides k. So the maximum power of p which divides this difference is the same in this range as when p k goes from 1 to p to the m minus 1. The maximum power of k which divides n minus k, and the maximum power of p which divides p to the m minus k, is precisely the maximum power of p which divides k. Because k is a number from 1 to p to the m, and so it will always, this is already the highest power of m of p which is possible, so this is this. So just say it again. So the maximum power, maximum, and I have written it before, but it's a slightly strange way, the maximum power of p which divides n minus k. So for k, k is an element from 0 to p to the m minus 1. So the maximum power which divides this is equal to the maximum power of p which divides k. And this is also equal to the maximum power of p which divides p to the m minus k. So we see that in this thing, whenever something above is divisible by some power of p, the thing which lies is directly below it is divisible by the same power of p.  And so it's not divisible by it. So the whole thing does p does not divide n. So now let's try to start the proof of the theorem. This is done in a strange trick which is quite remarkable. I don't know how one can come up with it. So we take, so obviously somehow you can want to apply this. So we take s to be the set of all subsets m of g. Such that m has precisely p to the m elements. So m is a subset, not a subgroup of g of order with p to the m elements. We know that the number n we have introduced here is just equal to the number of elements in s. So now we want to, we let g act on it by left multiplication. So g acts on s via left multiplication. So we have per g in g and some m is mapped to g m. Just every element in g, which as you know is a set of all g m in a, obviously left multiplication is a bijection. So it respects the number of elements. So it acts on the set s. And we can, so we have this action. So g can be decomposed into disjoint orbits. So thus, so we can decompose s into disjoint orbits, into the orbits, just like the orbits, with respect to this action. So we have s is a union of disjoint unions, a union of these disjoint orbits. So we have the number of elements and s is the sum of the number of elements in the orbit. So we have number n, which was s, is equal to the sum over all orbits for this action, which the orbit I call o, the number of elements in o, orbits for left multiplication. We know that p does not divide n. That was after the lambda. p does not divide n. So in particular, so if for all orbits, the number of elements in the orbit would be divisible by p, then p would have to divide n. So there must be some orbit such that the number of elements is not divisible by p. So thus, there is an orbit o, which is the orbit of some subset m, such that the number of elements that p does not divide the number of elements in the orbit. I'm confused. OK, so proposition above. Yeah, maybe you can remind me which proposition that is. I say the number of elements in this stabilizer of m is a power of p. Is that obvious to you? It should be, even though it's not in the moment obvious to me. Question is which proposition that is. This is the wrong one. This is not so the wrong one. So maybe as I, in the moment, well, I should at least, what, how much time do I have? We make one attempt to find out what the statement is. So does somebody remember which proposition I'm using here? Can make this an exam question. So who is smarter than me? OK, so yeah, obviously I don't have my notes, so I cannot look for it. So anyway, so I think I will have to leave this for the moment and I will tell you next time why the order of this. It's a bit annoying. Wow, I don't even have the notes. No, I cannot. What? What does the proposition say? No, what does it say? Yeah? OK, so that's the point. Yeah, obviously I should have remembered that. OK, thank you. So the statement is that, so if we have the proposition, which all of you remember very well, was that if G acts on itself by left multiplication, then the order and M is some element. Is a subset of G. Then the order of the stabilizer of the subset divides the number of elements in M. OK? So that was obviously the result we used, because we are looking here at subsets M, whose order is P to the M. So we must have that the order of the stabilizer divides P to the M, so it is a power of P. Thank you. So you managed to help me out. I hope I have still my notes. So now we apply the orbit stabilizer theorem, which says that the number of elements in G, which we know to be P to the M times R, is equal to the number of elements in the stabilizer times the number of elements in the orbit. Now, we know that P does not divide the number of elements in the orbit. So as P does not divide the number of elements in GM, because this was the number N, which we have found not to be divisible by P, it follows that as this thing. So on the other hand, this must be a number of elements. So if the number of elements here is a power of P, it must be the highest possible power of P. It must be P to the M. It follows the number of elements in GM is equal to P to the R. Now, the stabilizer is always a subgroup. So it P to the M, obviously. So it follows that GM is a P0 subgroup of G. It's a bit. It's a very funny proof to construct this group in this very roundabout way. But anyway, it's not supposed to be an easy result. So one has to have a kind of strange shape. It's somehow rather sneaky. I don't know how. Anyway, so it is. Also, you seem to be talking about something completely different all the time. And in the end, this group comes out. OK, so now we will try to prove the second C0 theorem. So maybe I hope you remember the statement, the second C0. So the statement was that if you have a subgroup K, and if you also have a P0 subgroup of the big group, then you can find a conjugate of this P0 subgroup that the intersection with K is a P0 subgroup of K. So let's try to do that. So let K be a subgroup of G and H, the P0 subgroup of G. We have to show that there exists a conjugate H prime equal to GHG to the minus 1 such that H prime intersected K is a P0 subgroup of K. OK, let's try to do that. So we look at the set of left cosets for H. So let S equal to G mod H be the set of left cosets. Obviously G acts transitively on the set on G mod H by left multiplication. You can just, if you start with H, 1 times H, you can get every G times H by multiplying by some element in G. And so just G times A times H is equal to G H. That's the action. And the stabilizer for this action of the element 1 times H, which is just H, is H itself. If you multiply an element in H with something which is not in H, you get outside H. And if you multiply it with something in H, as H is a subgroup, you stay in H. So the stabilizer of this thing for this action is precisely H. And you can see from this quite easily. So see that the stabilizer, if I take the coset, A H is precisely A H A to the minus 1. Now, you can easily see that if you multiply an element in A H with A H E to the minus 1, you are again in A H. And it's easy to see that it's the only way how you can do that. I mean, anyway, it's easy to see that the stabilizers, if you move the element, the corresponding stabilizers are conjugate to each other. Anyway, so this, if you want is an exercise, but it's kind of obvious. Now we want to restrict this action to the subgroup K of G. So I mean the group that operates is no longer G, but K. So the action is now from K times G mod H to G mod H. K times A H is K. And we want to decompose G mod H into orbits for this action, into K orbits. Before, if we took the G action on this thing, it's transitive, so there's only one G orbit. But now we take a smaller group. There might be more orbits. So as H is supposed to be a P-cellular subgroup, so it follows that the number of elements in H is P to the m. So the number of elements in G mod R, G mod H is therefore the corresponding number of elements in G divided by P to the m. So this is not divisible by P. It's actually, you know that, so as we know, that the number S, number of elements in G mod H, is prime to P. So it's not divisible by P. Because after all, we had that the number of elements in G is equal to P to the m times R. The number of elements in H is equal to P to the m. And by the whatever Lagrange theorem, we have that the number of elements in G mod H is equal to R, which is not divisible by P. So therefore, it's kind of the same trick as before. So therefore, we decompose it into orbits. We have again that the number of elements in S is equal to sum of the elements in the orbits. And so therefore, there must be one orbit such that the number of elements is not divisible by P. K orbit, so K times A H, such that number of elements in K times A H is prime to P, not divisible. So we put H prime equal to A H A to the minus 1. We have seen here that this is precisely the stabilizer in G of this A H. So then H prime is the stabilizer equal to G A H. It's a stabilizer in G. And now instead, we look at the stabilizer for the K action. But a stabilizer is an element which does not move this A H. If we take the stabilizer inside the subgroup, it's precisely the set of all elements in the subgroup which are in the stabilizer here. Because it's still the elements which do not move it. So it follows that the stabilizer, so I could call this for the action of K, A H is H prime intersected K. And again, by the orbit stabilizer theorem, we have that the number of elements in the orbit times the number of elements in. So we have that the number of elements in K times A H times the number of elements in the stabilizer is equal to the number of elements in the group, which acts, which in this case is K. But we know that the number of elements in the orbit is prime to P. P does not divide the number of elements in K of A H in the orbit here. So the highest power of P which divides the number of elements in K will divide the number of elements in the stabilizer. So if we write the number of elements in K equal to, say, P to the S times L, then P to the S divides the number of elements in the stabilizer. On the other hand, we know that H prime is a P group. After all, it is a P subgroup of G. So the number of elements is a power of P. It has a P to the M element. And this stabilizer is a subgroup of H prime. So it means that the number of elements in this thing is divisible by P is a power of P. It's a divisor of P to the M. So we see that this thing is a P group whose number of elements is the maximal possible one for P group, the maximal power of P. So it is a P zero subgroup, K A H, which is equal to H prime intersected K is a P zero subgroup of K. And this was the statement of the second zero theorem. Maybe this time I went slightly over, so I will stop here.