It's DDK and yesterday I told you guys that I was doing nothing but gig gaps But I do have a confession to make baby. Look, I gotta tell you guys what happened, right? So after I got done doing the gig apps running up that money, no cap Guess what? I did put a B&M for this low I did not honestly think I was gonna get it because it's over $2 a mile But we had to do we had to go well now this one is 68 empty miles 77 miles loaded so a hundred and forty five miles for 300 but the pull-up is so you who want it. Let's go and just like that. We are here at the pickup location It is so weird back here. I kind of got lost a little bit, but I got help me out. He told me that's actual address, so Hello, how you doing? I'd love to pick up from you guys to go to sturdily. Are you Yeah, I'm in the band. Okay, so you'll have to go back that way to your right. You'll see that Store back into there and we'll let him know it might be a little bit Okay, so it's over this way. All right, thank you You guys just got like three semis in all at one time. I did I came after I think I don't have to wait like two hours to get loaded up Oh Perfect time. That's on some ball. I was trying to hurry up and kids towards home before it started getting dark But I don't know I gotta go to start from Wisconsin with this So and we got everything strapped in it's time for us to get in and win It's usually I got the camera Come right here come right here Yes cars and crib cars and crib the best thing on YouTube to check it out And just like that we are here at the drop-off location. You guys would not believe what we are at We are at the same place that we went to the last time. We couldn't get unloaded because This joint says money to Friday 6 a.m. To 2 30. Oh This one open it's right off Hello No van I thought I was gonna be closed Yes, sir Okay, I'll meet you down there. I'm in a man. Okay. All right. Thank you No, I'm indicating culture. I work for myself. This is your band. You just take the contract You should do it Oh the number two get this money is what we love to do. We got one on free We about to get to this money. Oh, hey, we got one on free. We are picking up from Lowe's y'all know I'm gonna pull up and let's go get a pop like Crisco now We're picking up one item driving seven miles. The item is 24 by 24 by 60 It weighs a hundred pounds and we will receive a payment of 3402 get this money is what we love to do. Let's go just like that We are here at Lowe's to get this dough and skull get a pop like Crisco How you doing I have a freight order to pick up Yes, 9233 92348 what is hot water heater? Hmm. It's hot water heater All right, thank you. Yeah, you have a good one Actually stand it up or anything You're gonna put it right there and strap it in just like that we go. It's tough. Let's go get it on Let's go And just like that we are here at the drop-off location, baby I pulled off the dolly already and I looked at about a front door It was on the side door and it said leave the deliveries in the breezeway So I'm just gonna leave it on the breezeway and keep on moving and grooving in this money. Hopefully it's soon This dolly works way better than the old dolly. I had angle law the wheels just feel so smooth on this joint Then we'll step back into for all your nose Mac Order number three angle lie the freight app is letting us go out here and get this cheese We have a another order from Lowe's on freight get to this money don't hate We are picking up two items the dimensions are one by 48 by 96 I think it may be plywood, but I'm not sure we're gonna drive six miles And we will receive a payment of 30 402 guess what one this money up is what we're gonna continue to do Let's go just like that. We are here at Lowe's ski so we can get this dough G Hello, hey, don't I have a freight order to pick up?  Yeah, they told me to come here. All right Then they're gonna break it down here. Okay, got you. Thank you. I think it's two pieces of plywood I'm not sure because the dimensions are One by 48 by 96 and it's two Maybe it's this it has to be this. Oh, yeah, this is this it right here. Yeah Appreciate it boss. No, I said, you know You have a couple and I appreciate your help. I appreciate you helping me because I would have been standing here for 3,000 years 3,000 years Everybody's And We are here at the drop off.  See what they want. I do not want to put it in front of this garage door But they might have to go to work or something. I don't know so we're gonna see if not it is what it is How you doing sir? I'm here with your lowest order Yeah, so I'm not sure exactly where you want me to put it in. I didn't want to put it in front of you Okay, thank you. Just want me to sit it here No Yes You got it for you. Okay. How about them?  It's freezing out here today It's like 31 degrees. It's gonna be like the fourthest halloween this year. Yeah, they said it's supposed to snow tomorrow too, so Hopefully not Let's go. Thank you very much. You have a great one. How you doing? Sorry. I'm Jerome and that's Stephanie I'm living the dream. I woke up this morning. I cannot complain. I'm gonna grab a dolly They have the Okay, do you think that we need the card or anything or Okay, so it's all in one piece. Oh, it's two pieces Y'all see what's going on. Guess what? We about to go out here and get it done I know my nickname Rome. They're rocking on my microphone. Let's go Do you guys want to send it? Then it'll win So you want me to go this way before you went? Yes, it is. I'll have to kick it out some You mean like all the way over here I have to take a picture and we are good to go. Thank you very much. We have a great one And just like that. It is the end of the video. I hope you guys enjoyed the content we came We saw we conquered we kind of barkers, but I will say this It has been still very slow in my area as of lately I'm not sure what's going on. I need you guys to come up below and let me know How is it going in your area? Just so you guys know also that uh, the first order was on friday The second freight order was on saturday and then the third and fourth order was on monday, baby We got to that monday. So it has been going. So I don't know what's going on, baby Come on below and let me know what you think going on. I hope you guys enjoyed this video Again, if you're not subscribed make sure you subscribe Also, give me a thumbs up and don't forget to throw them a C's up If you're a part of the crew, baby, you're not part of the crew And you would like to be make sure you click that subscribe button Hit the like button and then throw them C's up in the comment section I'll see you guys on next one. We going to go before you already know this ddk and i'm on my way


Hey guys, what is up? Welcome to my channel. For today's video I'm going to share with you my newest pick up, the Hooda Beauty Rose Quartz Eyeshadow Palette. Well first of all this launch today you can get it online at Sephora and the Hooda Beauty website or double check but it seems that all of the Sephora's local to me do have this so they might have it in store for you. I forced my husband to take me out to an early lunch just so I could pick this up. That was my excuse but it sounds pretty. I'm so excited. If you want to see my thoughts on this then just keep watching. So today is November 1st which means my membership starts if you don't know what I'm talking about. I'm creating a membership for $2.99. It's called the Beauty Buffs. So if you join you will be a member of the Beauty Buffs and I'm just going to do one monthly tutorial on there. You guys vote. There's going to be member polls and then may or may not be some like giveaway. I don't know. We'll see. It's a fun time. So I just wanted to let you guys know if you were thinking about joining that it starts today. So all you have to do to join is just press the join button underneath this video and it will give you all of the details. You guys know there's nothing to make me giddy like a new eyeshadow palette and these make me excited every year because it's truly only once a year that Huda Beauty it comes out with an 18 pound palette. These are the best. Her 18 pan formulations and so is this 18 pans? 3, 6? Yes. So much better than her nine pans in my opinion and always, always chef's kiss on the color story. So I've been eyeing this when I saw it was at my loco Sephora. I had to scoop it up. It's beautiful. Anyways, ready ho. So let's get into the details of this. Like I said I did pick it up in stores. Here's the thing. I prefer shopping from Huda Beauty online. I feel like the products come very nicely packed and they fulfill orders super quick. However, I do encourage you to just buy this from the Sephora VIP sale no matter what tier you are. Just save yourself a little bit of money 10 to 20 percent. It's worth it because this baby is $67. That's a lot for a palette and they don't often go on sale. The older ones are at this point popping up on sale. But in the last two, three years of these palettes, they don't really go on sale. So get it at a discount while you can. It's worth the risk of shipping or just go to your Sephora store and get your discount. It's free to be a Sephora member. So if you're in the US, this is the Rose Quartz palette. Let me actually turn the light down so you can see the box. It is just a cardboard box. And if you do need to see the back, here is what the ingredients are and all of the information that you might need it to see. Now on the back says Made in Italy. So normally her best formula is Made in Italy. I know it's really dark and so you can see the colors of a palette and it has a 24 month shelf life. So here is what the packaging is going to look like. Absolutely beautiful. Honestly, this is my favorite cover that she's come out with to date. Like Naughty Nude was boring. Mercury Retrograde, not a fan of her eyes personally. New Nude, also her eyes. This looks luxe. It looks, it looks real sexy. I don't really like this. And then here is the back of the palette. Okay, you do have the mirror and here is the color story. And it's a gold toned, leaning, mauvey, purple, beauty-ness. I mean, can you tell I was excited for this palette? You can see that the component itself is kind of see-through. Can you see my hand behind here? But these are the colors. See, so like I said, it is $67. There's a matte finish, shimmer finish, metallic finish. And then we do have, we do have this one right here, which is weird. This product is vegan and cruelty free. It says that this palette offers 18 rosy tints and earthy shades across five different textures. We're going to get three high shine, buttery metallics, three pearly soft shine metallics, three quartz inspired multi-shimmer marble creams, and one pearl gloss hybrid. So while my lights are dim, I'm just going to go straight in and do the swatches for you. So we're going to start off with quartz. Ooh, that feels really good. Aura and gratitude. So these are all, why not all matte shades? This is what they look like up close. So quartz is a shimmery copper cream marble. So it's that marble shade, but it does have a nice soft shimmer to it. Aura is a earth brown matte, and then gratitude is a latte brown matte. Definitely leans a little bit pinky. Cosmic love, empowered and joy. Oh my gosh. Look at that. Cosmic love is a shimmery plum pink and purple cream marble. Oh my gosh. I am speechless. Empowered is a dark metal powder metallic. And then joy is a rose glow powder metallic. Okay, this should be interesting. We have precious energized. They feel so good. Okay, love stone. Here we go. So we have precious, a deep cool brown matte. Then we have blissful, a silver green creamy metallic. And then love stone, a white holographic and pink hybrid gloss. Cherish blissful and happiness. We have a cold pink matte, a silver green creamy metallic, and a bright pink matte. Mantra, serenity and abundance. This is a beautiful taupe. So we have a carbon gray matte that swatched really nice. A soft pink matte kind of has a little bit of plum to it if you ask me. And a dusty purple powder metallic.  I'm very happy. Final three, we have moon magic, radiate and self love. Some of these shades are just so eye catching.  And here are all of the shades. I mean, this palette looks beautiful with the exception of this weirdo right here. They are just beautiful. They swatch so nice. This is exactly what I was expecting the swatches to be like. So without further ado, I'm going to do this eye and then I'll be back to do this eye with you. Okay, if you get this palette, you have to do a look that's like this. I'm in love, in love. Okay, let me show you how I did it. Now, actually, you know what I'm going to do? I did get kind of a lot of fallout. So I'm going to take my Huda Beauty powder, which by the way, this is a recommendation for the Sephora sale that I have. It's the Easy Bake Loose Powder. It's my all time favorite powder. Well, except for the La Mer powder, but the La Mer powder is like $100, so this is my favorite powder price wise. So I have a little extra powder because I did get fallout from this and I'll tell you who, what, what, where, why. So I'm going to take my BK Up Beauty 202 brush and we're going into cherish. This is a very, very light pink and especially if you have like a medium to deep skin tone, it's going to be even more highlighting on you.  You can see it kind of just finishes up and brightens the area and then I'm just going to swirl a little bit in my inner corner. Very simple, very light. Refer number one brush and we're going into radiate. I'm going to use this as the inner half of my crease color and then I'm also going to bring it along the lower lash line like so and I'm going to go a tad deeper into its serenity and I'm just going to use the same brush. So this is a little bit more purpley. Honestly, these two shades, I find that they pull on this skin quite close. Like the palette could have done without having a both of these because you can see they're slightly different, but they're also like the same. So these two are a bit redundant, but they're so beautiful and they're so soft and blendable. I can't even complain because they just, they look so good. I'm using an elf $1 blending brush and we're going into mantra. This shade in credible, you guys. Colors like these can go so bad so quickly and not be blendable and be hard to work out. This color is so blendable and easy to work with. If you're a beginner with blending, like this is top notch quality. For a purple base, you don't come across shades that blend like this. So I'm honestly extremely impressed with this shade. It is this good for like no reason. For me to be happy with this shade, it did not need to blend as good as it does and it blends so good and you can kind of see I'm winging it out a little bit. I actually didn't do that over here, but I want to try it out to see if I prefer it. And then I actually want the lower lash line to be pretty smoky. I'm going to put a dark plum liner in my waterline. So I'm bringing it across the whole lower lash line. I can't beat this brush for a dollar. Look at that. I'm going to just blend the edges a little bit. Okay, so this is where things did get a little messy. I don't most recommend you use your fingers for all of the shimmer shades in here because they do get a little bit flaky, but I did want to just show you. So I'm going to use just this morphe shader brush and we're going into empowered. This is one that really has a lot of fallouts and makes a mess and you really don't want the shade to make a mess. I don't know if you can see but there's like purple particles everywhere because of the shade and it's a dark purple. So a finger applies this best. You can see I'm getting fallout. I'm going to get it on the tip of my shader brush and I want it to go right over the crease. This is why I said to apply a layer of powder down first and you won't get much fallout if you just use a finger and aren't stubborn like me. Pretty but messy. I'm just going to leave the fallout there until we're done with the other shimmer shades because work will need to be done with that as well. I'm going to go into a cosmic love. This shade, it's insane. This, I'm just putting it at the center of my lid. I'm still leaving the inner third bare but you can see we're building this galactic shimmery look and it's the perfect way to go from deep to light. I'm going to take a little bit of blissful. This looks more blue in person I would say. It is beautiful and I'm just taking a little bit and I'm putting it the inner quarter. We still have a quarter left. Okay this is a magical shade. This is moon magic and this is going to change everything in this look. It has a blue shift to it and I'm kind of just covering blissful all the way to the center of my lid so that the blue is going to pop against the middle shade and this makes the look. You can't tell me that this does not make the look because it makes the look. I'm taking the tip of an Esum W21 and we're going to go back into moon magic just because this is one of my favorite shades in this palette. It's literally a transformative shade. Absolutely beautiful. Last and final step I'm taking self-flub and using that as the inner corner color. I'm just using a morphe pencil brush just to add a little bit of glow and then underneath the eyebrow.  So here is the final look with lashes and all that notable product. I use the Huda Beauty Cheeky Vegan Blush Powder in Caring Coral and then on top I use the Flower Beauty Pyramids Cheek Colors, the highlights and these are perfect for the look. What a combo. Then on my lips I have the Huda Beauty Lipstick in Honey Bun. I probably could have gone with something a little bit cooler but anyways. So love this look. It is absolutely gorgeous. Love this palette. Getting into the more finer details of the palette. I immediately thought of Mercury Retrograde which was the palette from two years ago when this came out. I thought that they looked very similar. So here is Rose Quartz and then I'm going to put Mercury Retrograde next to it on camera. They look quite close and honestly I did swatch them side by side on my arms. So on the swatches right here you will see at the bottom is Rose Quartz and then the top are the closest shades I could find from Mercury Retrograde. So there's a number of shades that are definitely close. The Mercury Retrograde has a lot more different colors like there's blues, there's some different textures. Overall I think Mercury Retrograde is a little bit more fun of a palette in terms of having more colors. I think where the Rose Quartz stands out to me is that it has more options in depth. With Mercury Retrograde you had options for depth but I feel like there's even more options in the Rose Quartz and I think a lot of you will prefer that. Now do I think you need one over the other? If you have Mercury Retrograde I wouldn't say Rose Quartz is a must have. However I am happy to have both because I love them both and I can see myself pairing both together. If we could pull some of the blues from Mercury Retrograde into my Rose Quartz looks I think that they will definitely pair well together. So while there's a lot of similarities if you like the mauve purpley pinky tones in the Mercury Retrograde the Rose Quartz has those. The Rose Quartz just has more options within that color family.  I like how Mercury Retrograde has those fun blue pops and some glitter shades but I think in terms of what I would use more I would be more inclined to use Rose Quartz and Mercury Retrograde is my favorite hood of palettes and this is giving it a run for its money. So this one is so good but they are quite close. I definitely would like to have seen Hooda stay away from these mauve purple warm tones that she does a lot but I'm not complaining it's a really good palette. And then I'm just gonna show you Rose Quartz next to new nudes. This is a few years old but again you can see she really likes the purpley pinky tones. Rose Quartz is definitely much more cool so if you have new nudes I definitely think that these two make sense to have both. Rose Quartz is definitely the bridge between new nude and mercury retrograde. In terms of last year naughty nudes this is different if you ask me from Rose Quartz as well. Naughty nudes is deeper and it has a lot more warm tones so these two are not similar at all. Now one of the formulas that I didn't use that I know a lot of you are going to ask about is our little gel friend that's popped very close to the center. I took my um check it off because the yellow plaid was not vibing with this look. You guys honestly I'm just not a fan of this formula. I heard that the highlight version of this was pretty good but I don't like gloss on the eyes. It does crease this oil crease um it works good I would say as a base but for me this just isn't useful to me. I don't like it but when I do use it I use it as a base to put a powder over top so if you are using one of the more I don't know flaky shades it's going to grab on better and it also is going to help with the metallicness of it so I mean you can make it work but just know it is going to probably crease on your eyelids and it works best as a base in my opinion so I just didn't feel they'd need to use it for this look because it's just one of those colors that I never reach for and I do find to be a bit unnecessary in the palette so that's why I didn't use that one but let me back out. I did not do my hair today because I'm getting a haircut tomorrow and I'm just kind of letting it be but anyways final thoughts on this palette I really really love it with the exception of the gel shade there is not a dud in here and I'm not even saying the gel is a dud it's just not to my taste. The only thing I would say with application of these shades is that especially when you get towards and you'll feel it there's a few flakier formulas like this shade empowered is kind of a mess cosmic love and moon magic are a little bit more flaky so I do suggest using a finger to apply them or not doing your face makeup first. I love this, mattes are blendable, metallics are really shiny and gorgeous and multi-dimensional. Once again Huda Beauty has done it with this palette it is awesome. Every year she comes out with these. Every year I absolutely love them so big thumbs up in terms of whether or not I recommend you buy this. I would wait till this Sephora VIB sale if you are able to shop it just because it is a pricey palette. I do think it's worth it but it's especially worth it if you can get it at a discount and that's the best time to pick it up. So I hope you guys enjoyed this video and you found it helpful. If you want to subscribe to my channel already I would love it if you would get that or taking the time to do so and I will see you all in the next one. Bye guys, have a good one.


I like to feel as strong as a tree, calm as the sea, happy and free to move and breathe and just be me. It's time for a mookah moment. Hi friends, glad to have you here today. Thanks for joining my friend Teddy and I for a mookah moment. How are you feeling today Teddy? And you're feeling sad? You're feeling sad because you couldn't see your friends today? I understand. How are you feeling today? And remember, everything and anything you feel is okay. What matters is what you do with it. And you could always try a mookah moment. Teddy, are you able to try today?  First we move, then breathe, rest and sit. Come on Teddy, it's time. First we move, then we breathe and rest and sit. First we move, then we breathe and rest and sit. First we move, then we breathe and rest and sit. First we move. We're going to find easy pose. Easy pose, crisscross, applesauce, set up really tall. We're going to start at our head, roll your head around. That feels so good. You could go really slow. You could go a little bit quicker. And go the other way. Don't get too busy. And pause. Put your fingertips on your elbows or fingertips on your shoulders. And now you have chicken wings. Make circles with your chicken wings. This way we can try and go really fast. Or really slow, really slow.  Twist, twist, twist, twist, twist, twist, twist, twist, twist, twist, and then stretch out your arms.  Shake out your legs. Sounds like thunder. And then reach up to the sky.  You could tell yourself, I love you.  A little caterpillar wraps up in a cocoon, and a caterpillar turns into a... Do you see the clouds? Do you hear the birds? Wind shield white birds. And then it starts to rain. Swish, swish, swish, swish, swish, swish, swish. Come onto your hands. And this is cow. This, I feel this in my spine going down my back. I feel it in my shoulders. Nice job, friends. We're gonna come all the way down to the earth, all the way down. And go sleep. And wake up. These are really strong.  Try a really strong up dog. And go sleep. And wake up. And wake up. Let's try a down dog. Can you go upside down? I'm on my hands and my toes. I'm upside down. And up to the sky. Whoa. Whoa. Your leg. Whoa. Switch. Try the other leg again. You're so strong. This is a down dog. Upside down. And then we're gonna become a seed. Let's bury ourselves in the earth. Dig yourself a hole.  And I give you water. And sunshine. And we're growing, growing, growing. We're a tree. Can you try and stand on one foot? Whoa. Boris, that's okay. Smile when you fall down. It's so fun to try again. Whoa. Your arms are like branches. Reach them out far where maybe you're reaching up to the clouds. Whoa. Whoa.  Let's try the other leg. Switch. Whoa. It's fun to try new things. It's okay to fall down. Smile and try again. Tree pose. Whoa. We're gonna do our sunrise. Show me ready pose. Ready pose is hands at our heart. We're gonna reach up to the sky. Hello sun. And then we're gonna come all the way down to the earth. Step one foot back. And then the other all the way down. And then push up. Up dog. And then down dog. And then step your feet back to your hands. And we're gonna stand all the way back up to the sky. Hello sun. And ready pose. Let's do it again this time. Slow motion. Here we go. Reach up to the sky. Hello sun. And then come all the way down hands to the earth. Step your feet back and then come all the way down. Push up back up to your hands and then stand all the way up to the sky. Hello sun. And ready pose. Another one. Super slow. Reach up to the sky. Hello sun.  And then we're gonna come all the way down to the earth. Step one foot back. And then the other. And then as slow as you can. From all the way down. Push up. That's right. Up dog. And then down dog. Upside down. And bring your feet back to your hands. And then as slow as you can. Stand all the way back up. Reach up to the sky. Hello sun. And ready pose. Right into the next one. Reach up to the sky. Big reach. Reach even higher. Hello sun. And then coming all the way down to the earth. Step your feet back. Can you come down as slow as you can. Push up. Up dog. And then find back to your hands. Stand all the way up to the sky. Hello sun.  And back to ready pose. Shall we try one really fast? Okay take a breath to start. We do it again as fast as we can. Okay take a breath to start. One slow again. Ready pose. Reach up to the sun. And then come all the way down. Right foot. Left foot. Hold plank. And then come all the way down to the earth. Push up. Up dog. And then slow sun. And ready pose. We're gonna squat now. Come into a frog. Check your feet. We're gonna do a big frog jump. I want to make sure you're checking your feet so when we jump up we land right back where we started. Okay I know where my feet are. Here we go as big of a jump as you can. Three, two, one. Ribbit. One more. As big as you can. Check your feet. I'm still in the same spot. Here we go. Three, two, one. Down to the earth. We're gonna find ourselves on our backs. Let's find a happy baby. We're gonna hold on to our feet. Happy baby. What does a mad baby look like? We shake out your body. Mad being from side to side. Happy baby. And mad baby. Mad up to the sky.  Let's light up our candle again. Shake out your feet. That's the flame. It's on a birthday cake. Easy pose. Crisscross applesauce and sit really tall. If we sit like this it's gonna be really hard to breathe. So make sure you're sitting really tall and we're gonna take a deep breath together. This is like your belly. When you breathe into your belly it fills up like a balloon and then breathe out. Big breath in. Big breath out.  Big breath in. Big breath out. Ha. Let's do five deep breaths together. That's where I breathe. I feel a bit more calm now. It's time to rest. Squeeze your knees into your belly. Give yourself a hug. You could tell yourself, I love you. You're an ice cube. You're in the freezer and it's really cold. And then someone takes us out of the freezer. And we start to melt. We're melting into the earth. Melting into a puddle. Maybe we're on the beach and the sun is warm and the sand is soft. And you turn into a starfish. Stretch out your legs. Stretch out your arms. You're resting your whole body on the earth. I'm melting into a puddle. Look. I see the stars up above us, Teddy. Do you see them? Wow. Look at all the stars up there. Beautiful. I saw a shooting star. Do you see the stars? And we're going to rest our bodies here for five, four, three, and then wiggle your toes. Wiggle your fingers. And we're going to bring ourselves back up. Thanks for resting. It's time to sit. Find your easy pose. Criss-cross applesauce. At the end, we always sit. We might get really quiet. You might get really still. Try not to move. If it feels good, you could even close your eyes. I'm trying not to move, but I'm still moving. My belly's moving. Do you feel your belly moving? My tongue just moved. Do you feel your tongue moving? Eyes just blinked. Are your eyes moving? Toe just moved.  Let's try and sit as still as we can. We always finish our practice with a big thank you. Thank you in sign language looks like this. Chin to each other. Thank you. A big thank you. A thank you to yourself for showing up and trying your very best today. And a thank you to each other so we could practice here together. Thank you, Teddy, for practicing with me. And a thank you to the whole earth for all the lovely things that provides us. A big thank you. Thanks for practicing with me today, Teddy. How are you feeling now? You're feeling happier than when we first arrived. We know that all feelings don't last forever, and whatever you're feeling is okay. How are you feeling?  Thank you for practicing today.


Welcome back everyone our hunt for non-Abelian simple groups is almost out of close We know that a five is a non-Abelian simple group of order 60 And we claim that that's in fact the smallest order for which you can have a non-Abelian simple group We know that 60 is gonna work. There are three other contenders though that we have to throw out We've done every number from one through 60 and now we're left with the dreaded multiples of 12 We already took care of 12 mind you, but we're now left with 24 36 and 48 multiples of 12 are problematic for the following reason 12 just by itself Factors as two squared times three. So if you're looking at seal off two subgroups, you could have one or three Those so threes a possibility, but when you're looking at seal off three subgroups, you could have one two or four Remember this number has to be one mod three so four works there So when you look at 12 the way that two and three interact with each other It actually there's possibility or you can have multiple seal off Subgroups and therefore they're not necessarily normal So we have to look for other subgroups that are going to be guaranteed to be normal Of course if other primes come into play like five or seven you can interact with those But you'll notice that 24 36 and 48 they all have they all have the form Where we have a power of two We have a power of three and in particular we have at least two Two's and so thus getting a multiple 12.  Let's start with the smallest of them 24 24 factors as two cubes times three So if we start to try to count how many seal off subgroups could we have well like I said a moment ago with regard to two We could have one or three if you have one seal off two subgroup. It's normal We want to find simple groups. So we're not going to do that So you could then get three So we're gonna have Three seal off two subgroups Okay, so that's what this observation right here is now if we were considering the three groups, right? because we have we have eight your possibilities are one two Four and eight which of these numbers are going to be one mod three Well, you get one and four if you had a unique seal off subgroup of or of just one of them Then that would be normal. So really we're in the situation where we're considering we have three Subgroups of order two and excuse me of order eight and we have four subgroups of order three and in past videos You try to count elements and then show that oh you either counted too many elements or you have to have unique seal off subgroup 24 gets a little bit more confused in that regard. So we're gonna have to take a different approach to that Because after all one of the things hard about this one is that if you look at if you look at the two the seal off Two subgroups. What type of groups are there of order eight? Sure. There's some Abelian groups, you know z eight z four crosses z two The elementary Abelian group z two cubed, right? It could be one of those But you also have some non Abelian groups now you have like the dihedral group you have the quaternion group and so Which of them which one is it right? There's five possibilities just for those And you know if you look at D4 for example, it has five elements of order two It's got two elements of order four Q8 has six elements of order four one element for two Z two cubed has seven elements of order two z eight of course has two elements actually four elements of order eight and Two elements of order four is that right? You know and so it's like if we're trying to count elements of order two four eight It gets a lot harder because there's so many possibilities So we can't rely necessarily on the isomorphism type of the seal off to subgroup Which we've often done with our past examples because at least one of the seal off subgroups was cyclic I mean in this situation Because of the divisor three here We do know that the seal off three subgroups are gonna be cyclic of order three So we could try to count elements there like okay We have four of them, right? So there's gonna be four Subgroups they intersect trivially so they each offer an element of they offer three elements excuse me two elements of order three So we end up with four times two which is eight. So you have eight elements there but When it comes to the remaining ones, you know 24 take away eight is 12 You you might be like how can you have multiple seal off two subgroups if they're order eight? Well, their intersection could actually be quite large So it's a little harder counting elements, so I want to offer an alternative approach to dealing with 24 So what we're gonna do here is we are gonna take two distinct seal off two subgroups their groups of order eight I want you to consider their intersections. What could there? What could they be? So at a very important formula that we've proven in after down to our one is the following if I take the set HK which remember HK this is just the set where we take all the possible products of some h times some k Where h is in h and k is in k like so This is just a subset of the group. There are some situations for which this itself is a group But we don't necessarily know if that's happening here. We did prove that this set always Satisfies this relationship here that the card Nality of the set is equal to the fraction Where we take the order of the subgroup h times it by the order of the subgroup k and divide that by the order of The intersection of h and k so what we know is the following h and k is both Seal off to subgroups. They have order 8 so you get 8 times 8 which is 64 Okay, so what are the possibilities? Well HK is a subgroup that divide it's a subgroup of h So the possibilities for the order of h intersect k you're gonna get one two Four and eight well, they're distinct so it can't be eight Right because if h intersect k had order eight then it would be a to also be k that would make them equal That's not the case If your order was one right then we get 64 divided by one But this is this is sitting inside a group of order 24. That's too many elements. So you can't get one Right and if you did two then 64 divided by two is 32 which is Still bigger than 24. That's too many elements. So if the order of the intersection was less than two That'd be too many elements. So it turns out the intersection of any two seal off two subgroups has to be four in that situation and So Start this again in three two one As such We know Something about the size here. Okay, so this intersection is exactly four and so then when we consider the size of HK Well, sure, you're gonna get 64 divided by four. That's equal to 16. That does fit inside of 24 So that's not a problem here. I'm gonna I'm gonna change gears a little bit I'm gonna then look at h intersect k as a subgroup of h h has order eight H intersect k has order four. So the index is two The same is also true if you look at h intersect k as a subgroup of k its index in k is equal to two Now we've proven previously that every group of index two So I should say every subgroup of index two is normal inside of that group Therefore eights intersect k is a normal subgroup of h and it's also a normal subgroup of k I'm not saying h intersect k is a normal subgroup of g But it is a normal subgroup in h. It's a normal subgroup in k. And so now I want to look at the normalizer I want to look at the normalizer of h intersect k inside of g right because what's the normalizer again? the normalizer is gonna be the set of all elements g inside of g such that g H intersect k is equal to h intersect k g in other words The normalizer is a set of elements of the group that normalizes the the group in question right here the subgroup in question In particular there are some things we always know which are inside the normalizer the normalizer always includes the group in question It always includes The center of the group because that centralizes everything does it normalizes everything And so typically if there are if there are central elements that means the normalizer grew It's not just h intersect k But in general we don't actually know what the centralizer of this group looks like at the moment But what we can say is that the normalizer is going to be the largest It's the largest subgroup It's the largest subgroup of g for which h intersect k is normal inside of it What that means for us here is that if we find a subgroup of g because these are both subgroups of g If we find a subgroup of g for which h intersect k is normal in that subgroup That means that subgroup belongs to the normalizer So the normalizer of h intersect k contains h it contains k Therefore it takes it also contains any combination of elements from h and k in particular the set H intersect k is inside the normalizer as we observed earlier the The set because we don't even know if it's a subgroup the set h intersect k contain 16 elements All right, which means that the order of the subgroup n of h intersect k is at least 16 So the order of n h intersect k here It's at least 16 But we also know that this order by Lagrange's theorem because this is a subgroup of g This order divides the order of g which is 24. All right, so we need a divisor of 24. That's at least 16 right and The only number that's gonna work by Lagrange's theorem is 24 So we actually get that the order of The normalizer is 24 that means since the normalizer is the order 24 That means the normalizer of h intersect k is in fact g Which tells us that interest h intersect g is normal inside of g So we found a subgroup of order 4 that's normal and g So while we couldn't guarantee the seal off two subgroups were normal the intersection of any two seal off Two subgroups isn't necessarily normal inside this group So this exercise illustrates the power of taking normalizers using normalizers and Intersections of seal off subgroups can be useful to find a normal subgroup therefore. There is no group of order 24, which is simple So 24 is now off our list. Let's let's move and look at the last two here I guess I should have 60 here. We know 60 is gonna work So I'm really considering what are the numbers that don't work, but I'll put 60 back on the list Sure, we know 60 is gonna work. What about 36 or 48? Let's now tackle 48 Okay 48 has the form two to the fourth times three So the same thing kind of happens here. What are our possibilities? We get that inch in sub two How many how many two subgroups do we have? It's either one or three if it's one then We're not simple because we have a normal seal off subgroup So it's gotta be three if we're looking for a simple one if you're looking for three There's actually a couple more possibilities now devisors of 16 or two four eight One two four eight and sixteen Two doesn't work eight doesn't work, but you could have four you could have sixteen having sixteen is a lot You're gonna figure out that doesn't work, but four is kind of in this Goldilocks zone. It's not too hot It's not too cold. It's not too big. It's not too small We could perhaps get away with four seal off three subgroups and so by counting just elements It's gonna get problematic Now we could look at the normalizer at the intersection of seal off Subgroups just like we did with 24 that same argument would work right here by changing the appropriate parts And I'll leave it as an exercise to the viewer to do exactly that What I want to do right now is actually provide an alternative argument That would then that's useful, right? So there's you know, it's not just a one a one horse One trick pony. I think that the phrase goes right The not one maneuver takes care of all of these these ad hoc arguments depend on the factorization So while the previous argument does work in 48 I want to give us another argument to show why there cannot be a simple group of order 48 Like I mentioned just above when we consider the seal off two subgroups There's either one or three of them by the third theorem of seal off if it were equal to one Then this problem would be trivial because you'd have a unique normal seal off two subgroup So clearly we have to consider the possibility where the number of seal off two subgroups is three Okay, now we've used seal off threes We've used the third seal off theorem so much when we do these calculations We also are using the first one a lot because we that gives us the existence of p-subgroups in particular seal off p-subgroups Number two often gets neglected here But I should mention we're using all the time because we have only one seal off p-subgroup That has to be normal. That's a consequence of the second theorem because they're conjugates of each other. Okay So what I want to do is now consider that conjugation action and We're going to treat it much more importantly right now because we have three seal off two subgroups G acts on the Seal off two subgroups by conjugation that seal off second theorem But that that action has to be non trivial because they're all conjugates of each other And so if we consider the conjugation action of G on the set of seal off two subgroups call that X for a moment this is a group action and by Applying the strong Kayleigh's theorem that actually provides for us a homomorphism from the group G Into the symmetric group on X But there's three seal off three subgroups here And so this sub this symmetric group sx is essentially just s3 up to isomorphism Now this map right here cannot be trivial, right? The orbit structure cannot just be like the first subgroup the second subgroup the third subgroup That's an option Not for this one though because they they have to be all together the orbit for us the seal off subgroups is all three of them together Like so so we don't have a trivial homomorphism We don't have a trivial action. This tells us that the kernel of this homomorphism is not all of G Now let me let me kind of give you a hint on where we're going here If you have a group homomorphism its kernel is always a normal subgroup I'm trying to show that we're not simple every group has normal subgroups Everyone because you have the trivial subgroup is normal and the whole group is normal So that when you're trying to show that something's not simple you have to produce a non-trivial Proper normal subgroup aha, so I'm gonna try to argue that the kernel of this Homomorphism is a non-trivial proper normal subgroup. So it's proper, right? It's not all of G Why is it not trivial? Okay. Well, like I said a moment ago sx here is really just s3 uptie Symmorphism in particular the order of this group is six three factorial and So as we map as we map G into a group of order six the image of this group Has to divide six So in particular this can't be a one-to-one map if the kernel was trivial That would mean the map is one-to-one in which case then there's an isomorphic copy of G inside of sx But we can't isomorphically put a group of 48 inside of a group of order six. It's too small The co-domain is therefore some things have to map to the same place the pigeonhole principle here And as such in particular the identity is gonna have there's gonna be more than one element that maps onto the identity The kernel of this map is in fact Tribule excuse me. It's non-trivial. So the kernel is not everything. It's not nothing It's something and so therefore the kernel of this homomorphism is a non-trivial proper normal subgroup and hence G Is not simple. I really like this argument using the group action here The conjugation action of the group on its seal off in this case seal off two subgroups Because you can play around with this when the index is small like, you know The symmetric group in general gets really really really big, right? But if you look at small symmetric groups like s3 has only order 6 s4 has only S4 only has order 24. So if the number of seal off subgroups is small You can play around with the kernel of this conjugation action and you can make something work with that as well So I leave it as a I leave it as an exercise to the viewer here to convince yourself You can do the same thing with 36 that if you look at the conjugation action The kernel of that conjugation action has to be a proper non-trivial subgroup, which is normal And therefore there are no subgroups of order 36 either. So that then finishes our list, right? using these conjugation actions or by using normalizers we then can take out 24 36 and 48 and so that leaves that the only the only Order left for a non-abelian simple group. It's gonna be 60 I should say we're only we're looking at possible orders less than 60 There of course are not a billion simple groups of order larger than 60 take a 6 for example It's order is larger than 60 But this then proves that you cannot have a non-abelian simple group of order less than 60 for which a 5 then fills the bill the You know the nature of course a vacuum, right? There's no argument that prevents 60 from working because it actually works. There is someone who took on 60 Now mentally there's no other Non-abelian simple group of order 60 other than a 5, but that's not an argument We're gonna provide in this video So this was a pretty fun journey we hunted through and looked at every possible order now We can prove that there is no non-abelian simple group of order less than 60 60 being obtained by the alternating group before we end this video I want to present to you the following super awesome theorem that is way beyond our ability to prove With regard to what we've developed right now in this lecture series here So this is often referred to as the odd order theorem. It's due to Fett and Thompson I hope I spelled their names correctly if not tell me in the comments Anyways, it tells us that every finite simple non-abelian group must have even order things like 60 Which is an even number. You can't have an odd ordered Simple group and be non-abelian the only ones available are the cyclic ones of prime order and so the fact that Even orders are a lot harder than odds. It's sort of a very awesome very curious thing I'm presenting it on the screen right now because it's relevant to our discussion of simple non-abelian groups that we considered in the previous Well in this video and all the previous ones as well I'm presenting it for its sake of appeal But I want you to be aware that the proof of this goes way beyond the techniques We've been using with the seal off theorems and and normalizers and group actions and such It's way beyond the scope of where we are right now. So if you're a student trying to prove a problem about finite simple non-abelian groups They're not existence or something like in a homework problem or an exam I would highly encourage you not to use this theorem because your professor probably won't think it's very acceptable, right? I mean since it's if I'm your professor, then I wouldn't let you do it because we haven't earned this one yet Our mathematical theory hasn't developed this far yet. It's a cool theorem. Just want to throw it out to there just spaz you but Be aware it's off limits when it comes to proving, you know, homework questions and like a undergraduate abstract underclass or like a first-year Graduate level abstract underclass because it's just it's it's awesome, but it's just too hard for we are where we are right now So with that, you know, thanks to make it at the end of this journey for our hunt for Non-Abelian simple groups turns out a five was the smallest one there was if you learn things about finite simple Groups in this video or any of the videos in these lectures, right? Lectures 10 and 9 actually are together as a two-part lecture If you learned anything give those videos a like subscribe to the channel if you want to see If you want to see more videos like this in the future And as always if you have any questions about anything you see in my videos Feel free to post your questions in the comments below and I'll answer them as soon as I can have a good day everyone. Bye


From their curious writings made upon cuneiform blocks, there are endless areas of intrigue when it comes to ancient Mesopotamia, a fascinating and rare civilization which had an equally striking appearance, often adorned with trinkets, with tightly braided, often thick flowing hair, with royals regularly depicted as giants. It is also a very special area of interest for our so-called fringe research. The reason for this is that Mesopotamia is one of those rare chapters of ancient civilization – which, regardless of all previously noted, has strangely continued to be accepted by mainstream institutions, field studies apparently still flowing. As previously mentioned, this astonishing – and we feel – far older than currently claimed civilization is drenched with marvels of seemingly impossible ancient craftsmanship – many of which near impossible to explain in regards to currently claimed history. The reoccurring theme one finds when another post-Ice Age technologically regressed ancestor moves in to utilize these structures offered safety, will in turn leave behind an archaeological timeline. This then allows for an inaccurate and often blatantly ignorant dating. To muddy said waters are then met with a detailed, competent reconstruction of said lifestyles, religious beliefs, systems, etc., etc., all in regards to a permitted ancestor, rather than any details or logical explanation as to their technologies or constructions. However, as mentioned, going back to the recurring event we notice is the briefest of these supposed builders' legacies, for when one has laid claim to an antediluvian wonder, the lack of understandings regarded the fortress' strength, or indeed how to efficiently use them. The ingenious design of some of the most impressive fortresses of Peru, Sacsayhuaman, Kulap, for example, we poset if under the control of the original constructors would have been near impossible to evade and were completely self-sustained. Yet the academically-claimed builders all seemed to conveniently fold within less than a few centuries at most. However, the subject of most importance, and currently the most compelling exhibits of an ancient advanced civilization, is the nature of many of the artifacts either recovered or now documented as having been depicted across much of their stone-cut artwork. And across Mesopotamia, notably the Assyrian civilization, they had achieved levels of technological sophistication simply impossible to have achieved in the brief, currently-attested chronological life of said civilizations. Whether the Assyrian civilization and many others spanning ancient Mesopotamia have indeed been accurately identified, then an explanation for the array of remarkable technologies they had developed becomes a very hard area of archaeology to describe. Scuba divers, secret teachings, sophisticated levels and practices of law and healthcare, and most notably, and indeed the most vital section of the civilization's skill set, their intimate understandings that lay within their ability to create irrigation and agricultural systems which rival even those of the modern day. These tremendous abilities tend to make us suspect that either the dating of Mesopotamia is drastically off, or these feats of engineering were like many others adopted by this later settlement, ultimately decoded and claimed as an invention of their own. Astonishing legends of the past, accompanied by an astonishing level of sophisticated astronomical knowledge, is another crucial factor which not only indicates what we are attesting, but what we feel could have only come from an extremely old source, tributes to which seemingly found incorporated into nearly all surviving relics, yet, as if academia claim, this ancient civilization merely wielded stone and very later bronze tools. The question is, how did they create such astonishing ancient ruins? The Multiton Lamosu, a mysterious stone-winged horse we have covered previously on numerous occasions, it seems just like that of the so-called pre-Incas, displayed levels of sophistication specifically around horticulture, far in advance of what we should have logically presumed to see. It is as if they had a helping hand, by a far more ancient yet highly advanced intellect somewhere within antiquity. Are these uparts surviving remnants, memories left by a precataclysmic civilization, is capable of such sophisticated irrigating and building on steep mountain land with ease, we can for now only hypothesize. It is a pursuit we find highly compelling.