Mirror hey nigma. How's it going? Okay, so we're purging him on the opener. This is okay I think yeah, so the feral is the one with grace. I believe yeah But the priest gets cloned so we can only purge the feral, but this is fine. We get everything off now It's gonna be easier to get prox off We're in bash so like we have two options here We have the option to like line grace ourselves protect the field would maybe put like a palm up on the druid and Then once we have grace then try and push on the priest for dispels his PI is probably running out by then or We can trade dispels into him knowing that he has to fill it up Okay, we actually win a fear duel. That's okay as well If the priest is better, he's not getting feared there So like I see the penance right and he's on global basically and drop the fear on him That's fine, but yeah, if the priest is better, we don't get that So then being more mindful, I think generally protecting the field more when we can is Good rather than just rushing for the dispels and gambling Feral trinkets clone. We're in clone. We used to have an instincts. We still have the field. It's nice For all those with the charge, maybe I don't give the line to the feral there. Let's see Maybe I need to be a bit more careful when my feral's in clones. So just line feral as much as possible See like a guy. I had there's no reason to do this. This just risks me getting charged into bash And eating burns, but I don't don't think he has about right now. So Still he gets the charge into fear right? This is unnecessary. There's no reason to take this fear I can just line the feral for a little bit while my guys in clone and then go out afterwards And he's not gonna charge me and then we're gonna know we won't lose momentum here priest is gonna get a burn from this as well So definitely something to be more mindful of we we're basically in this state now where the feral's pops up and We're like a little bit behind here, which means we are gonna have a hard time purging He's gonna get a prox soon because we're still healing right? Okay, we charge charge or the flash into a bash. So the bash actually wasn't great from him because I'm already locked from the charge You could have bashed now, right?  I'm not sure actually Why don't oh I meld the burn, okay? That's fine Feral got unlucky. I think with his prox. You should have had a prox there somewhere We get the field off before getting feared so like If we go back, I don't I'm not gonna go back now, but Going back like 30 seconds plus like 35 seconds We are in this current state where I've taken two fears in a row without retaliation Because I'm gonna go back and show you what the error was here this what I'm doing here, right? This had an impact 30s at 30 plus seconds later Because I get charged here so then the priest can fear me off Because I don't have fear available. So there's nothing I can do to counter it Just because of momentum right like now now we eat a burn from it we're behind on momentum the feral has a proc I don't know if he's even gonna use that. I think he just doesn't use it or he healed himself or something So because we're behind obviously it gives him the charge available again To get me on cast so we're behind again have to use the meld still behind still behind still behind Having to cast a lot of heels unable to remove prox etc Fortunately get the field off that could have easily not come off, but we have to eat the fear for it because we're kind of constrained by positioning For all this continuing to momentum Going for the MD on innovate it's a risky one not worth too many buffs up Gambling So we're like we're still behind on momentum right all because of that early CC that they got on us due to that charge So just generally being mindful of the Okay, so this was a nice one We managed to dodge the fear there got charged out of line and bash But it didn't actually do anything to us because we had good positioning and awareness that you wanted to push for the fear So that was good But yeah, just if my boys in clone are just lying Don't I any sort of feral charge possibilities if I can obviously can we get this prop? Yeah, we get the proc. It's good. Oh Dodged another fear that was nice So we're kind of making a comeback with these fear dodgers, but like It's it's crazy like these fear dodgers are like like clutch plays right like in the moment Where if I don't do it, we probably lose the game, right? And it's crazy that I have to do two of those fear dodgers To recover that one mistake at the start of the game Where I Lied them to kind of build momentum as a result of it got sneaky burning him here.  I think he just dies. Yeah So we still weren't able to recover like let aside from anything that the feral's done on our team We still aren't able to recover in this game because of that early mistake and allowing them to kind of take control of the game and If we just assume that Our feral is not gonna do anything To outplay or take control of the game or whatever in return It's really important that we don't drop that early momentum to make it easy for him to just continue doing his his standard stuff and It be enough for us to control like the manner situation and the pressure and then outplay on fair dolls Because you see like the the third and no, what is it? Yeah, that's the We won the first one then he got second and third and then we've got fourth and fifth, right? And then he gets this sixth one, but it's off a charge again So this is a good child prefer so here we should technically here. I don't give the other feral enough credit I shouldn't go for this flash hill here. I should actually sidestep here and go for him I think this is a correct play there or or a palm No palms on cooldown For now anyway That's what let's let's break this death down a little bit more so here. We're kind of fine, right? We put a palm. I think that's correct.  I think that's fine Why do we two-ticket only? Let's see. I think three take three six. Oh, I think I'm scared of the fear.  He's gonna come fear me He might have actually gone for the fear of Fada Maybe not he shields right as I dodge So I could have gone gotten the three ticks away there, but I see why I didn't want to gamble it I see that we can start coming off as well. So I reshield That's okay. I don't know if it's a little slow. Let's see We could I think there's a little bit of dead global time here as well when we're dodging with the innovate We need to be more mindful of this because I think we could have put a renew up Somewhere while on global Yeah, like this that we could have easily put a renew up I think I'm also potentially holding global for his proc. So Again, it's not the worst. I think the main mistake is that flash he'll if we just We get we get the shield up on him Two tick penance. Pom is done. We could go for another pom. It's just about coming up now We could have gone for like a double fear that's really like risky though So yeah, the option is like to do one pom From here instead of this flash here because at this point we've been Like the pom would have been here, right? So the pom would have been up a while and we're gonna be almost So like at this point He's gonna have a shield and a pom still and we're gonna channel the him And we're gonna be getting to the first tick the priest is then gonna push around and we have to cancel after like two ticks and try and win the fiddle We have field available so that's doable. So I think that's the correct play here shield so like yeah the pom would have Would have been up and the globe would have been off about here So I could have been channeling now And I think that's enough to save him because look how slow he dies If I go directly for the him like as soon as I shield if I run out of line and him then The risky thing is if the priest is very good he runs straight here and denies the first tick Right. This is the issue The priest is good. He's he's gonna be very hit right here very quickly. So we get maybe one tick off Don't know if it's enough So I think that I think the pom into Into him is the correct play and then try to Usher the priest around this way so that they're together and then double fit Because we have peak our advantage right and then we can get the fair on him before he gets us


So over the weekend there have been some rumors circulating that Eric Wettel had received a couple of different contract offers from some different teams and one was even an AFC North team but he said that he had to turn it down but at the same time while he did turn those offers down it was said that he was actually still open to coming out of retirement to play and one of the teams that it was said that he was open to coming out of retirement to play for was one that he already played for a few years back that being the Baltimore Ravens and when Eric Wettel was here he was cool he's very outspoken but not outspoken in a bad way like one of our recent safeties was but he was outspoken in a great way he was a team leader he was the ultimate communicator on the Ravens defense he was the piece that really helped that unit jail and helped that unit click but when I heard about Eric Wettel possibly not minding returning to the Ravens I was cool on that I was like yeah I don't think that I don't think that would really be in their best interest because I know there were a lot of people that were hot on the idea but I was I was ice cold to it even ice cold in the coldest water and he already checked them out make sure you check out their website in the description so you can shop your coldest water using Graven Viz the code for 10% off and also you can get into their weekly giveaways they got so many giveaways every week man it's so easy to enter all you gotta do is put your email and that's it so that's a no-brainer but I also also think it's a no-brainer for the Ravens to pass on this and I don't expect there to be any interest from the Ravens but I don't think there is any reason for there to be any interest because while Eric Wettel was cool when he was here like I said he ultimate communicator he was the one relaying all the defensive calls to the team he was the one getting everybody lined up getting every everybody in position Eric Wettel did his thing when he was here but something else that we noticed every year that he played here was that his body continued to slow down he was smart smart smart smart smart he knew what the plays were he was good at dissecting the plays analyzing the plays and knowing exactly what was coming but his body it's body just couldn't keep up he was slowly breaking down slowly but he was breaking down nonetheless still was a solid player even in his last year but it just it was time it was time and again this is where Eric the Costa this is why he has been such a good GM here well actually he's been pretty great GM he's had some misses now so let's not act like Eric the Costa hit on every single thing that he's done he doesn't have some misses but his makes they outweigh the misses by far but one thing with Eric the Costa that he's shown to us as Ravens fans and just really to the NFL is that he don't care how much you liked in the locker room he respects it but that's not gonna change his decision on whether it's time for you to go if he feels your time is up and that you can no longer contribute to the Ravens and that you're no longer needed he will cut you he will cut you he will release you he will trade you he will get you up out of there and of course it's nothing personal well maybe with the last safety we had in mind a little personal but usually it's nothing personal it's strictly business and Eric the Costa keeps a strictly business and he usually does what's best for business and for Eric Weddo he could come back sure I'm sure he's still got a little bit left in him he can come back but what will he be a rotational guy would that be what he would do but even then what if while he's in on in on a rotation what if somebody's coming right at him because you know players in the NFL now a faster than ever and of course not everybody is a Tyreek Hill but every time I just whenever whenever I think about Eric Weddo I think about Tyreek Hill Eric Weddo one-on-one and a lot of guys will lose that match up but I just remember that match up specifically when Eric Weddo was one-on-one with Tyreek Hill Tyreek Hill was in the backfield Eric Weddo was just standing there Tyreek Hill looked at him he just ran right past he ran right past me again like I said I know Tyreek Hill would do that to a lot of people and he still does it to a lot of people but we've seen a countless number of plays with Eric Weddo just he couldn't do it anymore and then he went to the Rams and he did all right there he wasn't bad but even the Rams they said okay thanks a little but that's it that's it and they were even gonna he wasn't gonna be a starter anymore he was not gonna be a starter anymore so when they said that he wasn't gonna be a starter Eric Weddo said oh okay no I'm let me go ahead and go ahead and retire because I'm not gonna be a backup so with Eric Weddo and I know there's been also there have been talk about Tony Jefferson too about him possibly coming back to the Ravens and I just I don't feel like again both of those guys great for the locker room they will both be great for the locker room both were leaders both they were so cool with each other they're cool with everything you never hear a peep out of anybody talking about either one of those two as a bad teammate you never and to on top of that you hear so many great things about both of them as teammates you hear that man but I just don't feel like there would be a place for either one right now with the Ravens I just I don't I don't see it I don't the thing that the Ravens are missing right now and they have been shading it as much as they can of course they've been put Marlon Humphrey there a lot but the thing that they're missing is a slot cornerback and Eric Weddo he ain't covering the slot Tony Jefferson ain't covering the slot they could of course be nice depth as safety they'll be nice depth as safety but more so Tony Jefferson and Eric Weddo but box safety and I feel like at this point in his career Eric Weddo will be more he'll be better as a box safety even though I think that he would even struggle but there were some times where again it happens to everybody it's not just Eric Weddo but there was some time you get run over he'll be right there but he'll just get run over and like I said anybody it could happen to anybody but again his his body was just breaking down so I just I don't think a reunion needs to happen I don't think a reunion should happen even with Tony Jefferson man I am nice for the locker room before on the field I just I don't think it's a necessity it would be a big-time luxury if they were to bring Tony Jefferson back and then it's like hmm if you bring Tony Jefferson back who's out who's gone because he would take up a roster spot obviously so somebody would have to vacate their roster spot but if if Eric Weddo does end up coming back you know one thing to think about too because they said there was one AFC North team so I would think that it would be the Browns because they lost their safety Grand Delford and Eric Weddo is not I mean Grand Delford is obviously much younger than Eric Weddo so he's on the up and up and Eric Weddo is on the down and down but I would think that it was them but you really never know it could have been the Bengals could have been the Steelers you know still still is one of him before he they wanted it before Raven sign up Steelers had one of them but he ended up choosing the Ravens wow that seems like that was so long ago was that three four years ago think was four years ago something like that wow that's crazy Eric Weddo he well and and Eric Weddo the thing with him that I love is that he got so much respect from Ravens fans from the Ravens players from the Ravens front office he got crazy respect and like when you hear Ravens fans talk like it is crazy because with Eric Weddo he he only played for the Ravens what three years and when you hear Ravens fans talk about Eric Weddo you would think this dude was here for like six seven ten years man that's the the respect that this guy has man you would think that this dude played for the Ravens his entire career something like that I'm sure he probably he probably wishes he would had but not Eric Weddo help the Ravens get a Super Bowl man Eric Weddo he was part of the Raven Super Bowl run because and called bolder I think I think he ended up running oh no he had no Ray Rice I think Ray Rice ended up running over Eric Weddo in that play did he give Eric Weddo a stiff arm on that fourth and 29 that was actually short but we wasn't complaining but anyway cuz he was on that charges team well that was a nice comeback game too wow but anyway this this doesn't need to happen it has no reason to happen the only thing that he should come back to the Ravens for would be to coach but I'm sure he doesn't want to coach if he wants to play okay well go play but I just I don't think it would be a good fit for him with the Ravens right now because they they are trying to get faster they need to get a little smarter on defense they had some breakdowns and communications not last week but against the Chiefs so I think with them the biggest thing that he could come back for would be to be a motivator and somebody that could keep their energy up and somebody that could possibly help them just remain focused even when things get hard because that's one thing that's one area where the Ravens seem to lack is to remain focused when things get tough when things are not going in their favor that's where they're lacking that and if Eric Weddo gonna come back to the Ravens do that oh okay well hey bring them on bring them on every time that the Ravens are down by three or more points bring them down every time that the Ravens play the Chiefs bring them down when the Ravens in the playoffs cuz that's what we need so anyway team keep it clean I appreciate y'all I love y'all and I'm out


Are the European Central Bank keeping benchmark interest rates unchanged? Here now is Eric Weigan of U.S. Bank. So Eric, the ECB saying it expects interest rates to, quote, remain at their present levels for an extended period of time and well past the horizon of net asset purchases. That's a pretty significant development. You know, it is. But it is somewhat expected. There was views going into this meeting that there was going to be some moderation in the language. And effectively, that's what it is. They removed the downside bias. Previously, they had suggested that they would maintain at current or lower levels. So they removed the language regarding lower. And you know, as a dovish tone, they've issued language suggesting that it's going to be for an extended period of time. So in some ways, it's acknowledging the accelerating growth that we're seeing out of the eurozone, but also acknowledging that there remains very little inflationary pressures. It also means that negative interest rates are here to stay for at least the foreseeable future because those asset purchases are set to end in December of this year. Yeah. And they haven't addressed the asset, the level of asset purchases. They will continue at the 60 billion euro level per month. And because there was debate in the ECB over whether to get interest rates out of negative territory and then end the stimulus, but clearly the ECB throwing cold water on that idea. Correct. I think they'll, like we've seen with a lot of central banks, they'll tend to err on the side of caution, you know, preferring to extend accommodation as opposed to, you know, jeopardize what has been a fragile recovery. Now from a stock's point of view, are you more bullish on European equities or U.S. equities? Because a lot of the strategists I've been speaking to favor Europe over the U.S. You know, Europe and developed international has become more of a, you know, of a popular play. And we would echo that. We do think that we are seeing political risk receding in the eurozone. We are seeing improvements as far as the economic activity, whether you look at GDP or industrial production levels, they're improving. Unemployment while stubborn and elevated levels is coming down. We are seeing the consumer engaging once again. So we tend to favor equities, but we do acknowledge that, you know, Europe is more, you know, attractive at these levels. And it's also because the European economy seems to have more room to run. They're at the earlier stages of the economic cycle than perhaps we are here in the U.S. Earlier stages as well as continue to have more monetary accommodation being provided. So that continues to be supportive. Okay. All right. Well, Eric Weigand, thanks for joining us from U.S. Bank. I'm Scott Gam and you're watching The Street.


Hey there lovers and friends! I'm joined today by my official lover and my best friend. Are we best friends? Yeah. Do you consider me your best friend? You don't really hear you say that. I got like hoes, you know? Yeah. Probably hurt. You're my best friend. Yeah, you're my best friend. Yeah, that's nice. Well, this movie is a partnership sponsored, nope. I got all, I got all emotional over the best friend talk. This video is done in partnership by Netflix and that sounds really cool to say. I'm really proud to say that. So Netflix has a movie called Set It Up. It's a rom-com. If you guys know me, I love, I'm an aficionado. Is it right term? Aficionado? I think you should say like, I'm the sommelier of rom-coms. I'm the sommelier of rom-coms and I loved this movie so much. So when they reached out to me and were like, we want to partner with you, I was like, yes, I want to partner with you too. So the movie is out right now on Netflix. If you guys haven't seen it, here's a little trailer. I'm always the last one here. I'm always the last one here. Kirsten keeps a cotton in her office. I'm going to take a nap, wake me up at midnight. Rick keeps a temperpete cotton in his office. I will see you tomorrow. What are you talking about? You're coming with me back to the office. The two of them are always in this office. Let's just set them up and then we're free. Let's get it. You have no emails. The worlds are oyster. You use too many exclamation points. You have the romantic personality. The seventh grader. I feel like it'd be at least high school. To me, it's going to be a Netflix's classic rom-com film and there's a line, what was the line? We like someone because of the good things about them. We love someone despite all the things that we can't stand about them. So I thought it'd be fun for me and Jared to put together a list of things that we can't stand. I can stand it obviously, but like I low key can't stand it about you. You ready? Yeah. Okay. Can I go first? Yeah. I want to go first because I was actually hoping that you wouldn't do this thing that you never do. So I could say you never do it. You know what I'm going to say? Compliments. Jared never gives out compliment. No, you don't. This today, today I even set you up. I said, this is my natural hair. Like I don't have extensions in and you were like, oh, yeah, I know. And I was like, oh, how did you know?  That was an opportunity to compliment. That was a setup for a compliment. No, no. It's almost, it's a running joke at this point, but you know what's great about that? Compliments and words of affirmation are actually the bottom of my love language. But I do. Okay.  I do. But like super seldomly. These are all lies. These are all lies, guys. Try, try, try to give me a compliment right now. Try.  That's nice. Thank you. All right. Well, one of my things, one of my biggest pet peeves of her, she never washes her workout clothes. She always wears the same workout clothes over and over and over again to the point where a lady was on the treadmill at the same time right next to her and got off the treadmill. We all know why. Do that for you. What? Because you do the laundry. So I don't want to stack it up. I, I wash my workout clothes once a week. I don't never do it. I also do that for you. I also do that for you. Know that this, because it's actually right here. We should do a shot of it.  And so I don't want to fill it up. I love you. You're welcome. Okay. Acts like he's banned from shopping. In Jared's mind, he's not allowed to go to the stores. So he, he, he only talks about the things that we need and never purchases them. Just like, I think he thinks I'm the shopping genie. He says things and then is like, Oh man, we're out of soap. When we go, I go, we go together. I'll purchase them. You're afraid to go without me. We have to be together. That's a thing. You'll never pick yourself up. Like even, what did you suggest to the day that we needed the blanket thing? He's been making this suggestion for a year. We need different blankets. We need different blankets. No one has stopped you from buying this blanket. Nobody has said you couldn't purchase the blanket. I'm just saying, I'm just saying I've purchased things in the past and they did sit back. List one. Hmm. Right. Hmm. I can listen to my one. You told me that you never, made purchase that you're like, no, not that we had this argument at the day and he said millions of things. I said, name one. He said millions. I still care. They're there. What's one? She knows, she knows, she's running right now. She knows that.  Go ahead.  She doesn't dump it out and she doesn't even take out the tea bag. She leaves it in the cup and it's just sitting there. If it's not in the cup, it's in the sink. It's one of the other. But you saw that yesterday I reheated my tea at like 3pm. So there's days that happens, but she won't ever do that. Well, I have the intention to possibly have it later in the day. That's why. This is just like a general complaint, slacking on your kitchen duties. You're just not that great anymore at that. You used to be an all-star. I'm working. I'm all over this. I know, but there's like kitchen, there's chicken from yesterday right now. But there's nothing else on the counter. I was hoping that maybe Shannon put the food away. If there's going to be one gross thing, it's going to be chicken on the counter. I left it out. I thought you were going to eat it yesterday. It's not only the counter. It's in the pan. It doesn't matter. We just have chicken out. Just this chicken living with us right now. We're both busy, so we know this. Sometimes you could jump in the dishwasher. Sometimes I don't mind doing the dishes. I do them 90% of the time. 10% of the time you can jump in there. All right, let me go. Having all these secret places for your dirty clothes. They're never in the same place. They're never in the same place.  They're being a pile on the deck. It's when I do laundry after going to scavenging. And then she tries to come to me and be like, yo, you just did laundry. Why is there stuff in the basket? Because after I started it, it's probably a secret stash somewhere. Yeah, the behind the dresser. That one I stopped doing. I stopped doing it because I now put it on the secret pan. My last one that I'm going to put down is extreme rule following. Extreme rule following guys. Can't do that. Don't do that. Don't walk yet. So many rules in my life. Not yet. Be careful. Slow down. It's just like, bro, like, can we just... Like, she acts like I never break rules. What rule have you broken? Too many. Millions, right? You need to have gotten a person to go to a grocery store and like you're eating from the bulk barn thing is a problem with that. Yeah, but you're from Toronto. Like, you're going to J walk. I've seen people get thrown on the street in handcuffs through jaywalking. I've gotten a ticket. I've gotten messed up by a cop before jaywalking. I've never heard the story before. In Burbank I have. I was running to the bus. I told you this. No. I was running to the bus to get. I was late for work and this cop just cut me off and he made me take off my shirt because I had tattoos. Why? Because I live in America. This is America. We grew up two different ways. So yeah, now I follow rules, but it's not because I'm a goodie to choose. I'm not trying to get thrown around by the police. I'm sorry that happened to you.    That's not a fat book. Of things I didn't even say. You did not have things you didn't say. Oh yeah. Let me see. Let me see it. Read it. Just list it off. You don't have one more thing. This millions of things. Millions of things he didn't say. Millions. All black pages. All right guys.  So do what we call the set it up challenge. Why not? Go on your phone right now. The last person you text make a list of all the things that they do that's super annoying and then say at the end and yet is still messed with that person. I still adore them. And most importantly, if you have yet to see set it up yet, go to Netflix. I'm going to assume that majority of you have an account. If you don't, it's probably a good investment of your like under $10 a month. Go into the info box below, find all the information on how you can do that if you haven't already. And yeah, I'll see you next time. That's all I got to say. Thank you for coming. You too. That's a good exit.


Good afternoon. My name is Steven Smith. I'm the Director of IT Infrastructure at MCNA and welcome to OpenStack Summit in Austin, Texas. I'm going to be talking to you about getting OpenStack out of the corporate basement. So there's a lot of elements that you'll have to go through depending on whether what your current infrastructure is and where what your starting point So we're going to talk about components of a successful project proposal, which is where all projects start from Discovering and satisfying the corporate infrastructure requirements Addressing the upper management concerns and any hesitations that they might have promoting and presenting your OpenStack project team training Some no-cost proof of concept strategies and We'll review a little bit of a couple elements of some infrastructure design and finally alleviating risk so as all projects start you'll need to start with a proposal so the most important part of the start of starting such a project is Best to clearly state your business goals and objectives. So what are your objectives? Are you reducing your overall operating costs for servers? You want to run some additional applications? Will your investment pay off and what will your total cost of ownership be from your current solution? There are other elements that of OpenStack that you want to Also deploy do you want to enhance DevOps or deployment services deployments and other services? So What is the reason for going moving over to OpenStack? So You want an open flexible reliable technology base for the for the future? So MCNA is a dental insurance company. We have to be HIPAA compliant and In our in a private cloud you'll have vastly easier compliance since you have control over all your your infrastructure and as a Cases in some of the public cloud systems, they will charge you up lit costs For dedicated instances or resources So what are your current pain points? What is it you're trying to to solve? If you're in a public cloud system all the hardware is shared And if you'll notice on top, you'll see If not occasionally most of the time Stolen CPU at the top that means that those are CPU cycles that are being taken from your instance and given somebody else If one tend to just having a problem or an issue it affects you as well If you have support from your Cloud cloud provider We've been on hold for hours at a time during during major outages with no resolution Lot most private public cloud Providers you have absolutely no access to the console on boot up And which vastly reduces your Your your time to solve issues Tier one or tier three support a lot of Cloud providers, they're all the same Matter of fact most of our techs We don't even bother calling because we know more than they do and calling them again. They just Generally have no answers We have instance maintenance in the middle of the business day, which has never made sense to us whatsoever we had You know a reboot server or a stop in two o'clock in the afternoon For business continue a continuity and disaster recover An infrastructure in one site doesn't necessarily mean Doesn't necessarily mean that you have fail over in a second site and generally you'll have to double your costs to have fail over in another site In the case of your private cloud, it's just the cost of a second set of hardware Any Upgrades that they do hardware schedules or any decisions they make you're at their mercy and you have no control over them So those are our pain pain points, but what you have to do is in your proposal is list your find out what your pain points are and and Resolve them You'll want to hit on all the service improvements that that you have Total hardware control gives you faster response to resolve your issues Higher level of security you have way more control over the security and again if you're have to comply with some regulatory laws It makes it a lot easier You have a greater scope of instance and service customization to suit your business needs. You're not stuck with whatever resources they your cloud provider provides as a as a For resources for for CPU and memory You can design your computing environment for any level of performance you you want you can have different hardware for different parts of the infrastructure And your hardware network performance can be specified and customized so Your implementation phases are going to be very important Mostly you'll be starting off on a sandbox where you will have no resources where Maybe your boss isn't even even aware that you're you're going to propose this so You can use miscellaneous hardware you use dev stack Some of the open stack all-in-ones installations Virtual box and as we did we start off with a bunch of workstations that we just grabbed and and installed open stack on So your next proof of concept once you've gone through the sandbox phase your proof of concept There's other Resources that you can use there's some free trials out there that you can sign up for And you'll want to bring it up to the next level of a separate controller compute and and storage in your pilot a lot of hardware vendors will lend you Hardware once you're getting more serious about it Either in their own Facilities Or they unlikely they'll they'll ship you some but There's also some vendors that will do allow you to do hardware trying buys Where you purchase the hardware and if you don't like it you just return it And at this level you should purchase a subset of the hardware that you're going to use for your final production design and then Then you move on to your production environment where you'll actually Deploy the hardware your final production hardware and Start migrating your production instances So in each one of the phases You probably want to you want to be finding out does open stack play well is it going to work for your company? Is it going to solve? Your pain points and These are usually isolated Implementation tests and you want to prepare for your your your proof of concepts In the proof of concept This is where you will find out a little bit more if it'll really work and You'll actually Try to be testing the implementation of open stack on the separated separate compute storage and and Hardware Pilot and getting your feet wet You're getting the execution of the the the implementation framework you want to be exercising How you're actually going to be working with open stack and by this point you should have the whole production environment Designed Most important take your time. This will not happen overnight Could take four to six to eight months depending on approval processes and It's important to get each phase approved and accepted at each level So The first step towards a successful open stack deployment setting is realistic expectations. I Under promise an over deliver Cloud components will fail and you'll need to design and high availability and it will take longer than you think it will So set up a schedule up front and stick to it from from the very very first sandbox Try and get a schedule set up. It's all it goes all the way out to a production deployment Like I said, it will take longer than you think so be very conservative When you're choosing your dates and how much time it's going to take to do each step so You're going through your sandbox part way through the proof of concept and You're going to have to decide what are your infrastructure? Requirements so you want to determine the services that your current infrastructure provides whatever that may be and then You want to map them to the open-stack components. So these are the basic open-stack components That every cloud should should have and Which a lot of public cloud providers already provide so Whatever you decide or whatever your infrastructure currently is map them to The corresponding open-stack components What are you not using in a current infrastructure? What are you not using with a current? cloud provider If they were free to use would you start using them? So These also mapped a lot of cloud providers of offerings But perhaps you're not using them because You're not there. They cost too much or they don't fit within your budget. Well an open-stack. These are would all Be free to use as long as you have the hardware to support it which a lot of these will Basically work on a controller You may need to buy some extra hardware, but you can plan and budget for that. So One of the toughest parts is not the technical part. It's actually getting management approval and managers read Probably more than tech guys do and they know of they've read all the articles and They've read reports so We'll just go over some of the Some of the concerns and they might have so their first concern is the pool of candidates Open-stack is Pretty small at this point it's growing but They aren't standing out on the street corner with signs will work for four open-stack. So Um Security is another huge concern if you're building your own infrastructure where it's a security. How are you mitigating it? What are you doing about it? Your deployment, how are you going to deploy open-stack? Are you going to deploy it from by yourself? Are you going to get help? Are you? Wow, are you going to get help? Are you going to use some other tools? And Then why should we be using open-stack? Well, yeah, who paypal and some other large players are using it. Yeah, but we're a medium-sized company. So What can we do? We don't have all the resources of these? Well, I'll show you how we can find them later on What's the system stability of open-stack I've read it's not stable Well, I'm going to through my proof of concept. I'm going to show you that it is stable and I'm going to document everything to show you what what's possible and with open-stack and how stable it is and What if you don't have the capex this year for hardware well, there's other leasing options and other options that you can work out with a lot of the hardware vendors and Just don't Go telling your manager or your boss everything up a management everything. That's great about open-stack. Well Everything is great about open-stack. There are some Issues and there are some things that you won't be be able to do so make sure they understand both the good and the bad so you've got part way through your project and You Want to accelerate it? What can you do about it? Well, you can start seeking out stakeholders and collaborators and collaborators and innovators in your in your company Top one on my list was the dev team They wanted the ease of deployment for test servers They want to start moving in the docker. They wanted to Be able to launch farms at a click of a mouse Which is all possible with open-stack So you can get them on your side and seek out others then you can get some Supporters of your project So when you're promoting and keep concepts simple don't be too techie. Yes, you understand all the technical Requirements behind it, but your manager may not understand what you're talking about so Lot in the way you may get arguments as well Well, open-stack can't do this and open-stack can't do that or we can't you know, we're also we're looking for something a little different So present them all counter that with solutions well, yes, but we can do this or We have we can get around it some other way or do we really need it and again? I'll go back to what is your current infrastructure and what are you using? You may have all Kinds of paperwork and design drafts and and everything else But somebody's gonna read it at some time. That's an executive. He's gonna say what is this? I so at the beginning of every plan I wrote executive briefs two three pages long that Basically outline all the technical details Before below and that was in the document We gave many demos We had one demo go bad and it actually did more damage than anything else so make sure you practice them and and Make sure that they go go smoothly because it's It's one of the worst things that can happen if they actually see that something's broken or doesn't work properly If you don't talk in the correct terms of management, they'll hear clinginese. They Won't even know what's what's going on and stay away. So let me make sure you stay away from the technical jargon And Speak in terms of business and goals and metrics That's what they understand. They don't understand That it's really cool to have new hardware or you're running you run 80 cores on a single chassis or anything like that and As it doesn't really have a return on investment think more of TCO toll cost of Ownership you're gonna. Are you gonna reduce that hopefully and? That'll be something that they'll be very interested in If you're gonna start stating savings and calculations in your project plans make sure That they are accurate be very conservative of them because if you get into one phase or you get into Production and they're not realizing the savings that you said that you were gonna get it'll be viewed as a failure and Don't make promises that opens that can't keep I've had guys in meetings and demos. Yeah. Yeah Opens that can do that No, it can't really and then you get in trouble because it could be a component that one of the managers will Latch on to and so where is that? I don't see that so part of Part of the process you're going to have to have people to support it design it and building it build it so How are you gonna do that? Well, they can start reading the documentation which is a concept. I know foreign to most techie people, but The Open-stack docs have gotten much better over over the years and they're very good now a Lot of the installers deployment tools have documentation on their website that you can get a lot of information from from and Build a build a lab for you guys. Let them let them fool around in it It's nothing better than typing away on the CLI and understanding, you know, what does what? It's also inexpensive training materials books ebooks at Linux Academy also has some Training materials that they start offering. There's lots of expensive training to You can either bring in a trainer or fly him out to a Training facility Make sure you allocate time for system ins to play Make sure they're not busy all the time so that they can actually progress in their learning and Get to know and as more that they learn the more that they'll be eager to to start implementing And for that matter you can create a training schedule as well summit videos for past Open-stack summits awesome source of information they can watch them at lunch or even let them watch them during the day and You can also explain them Open-stack would be good for your career. You're not getting replaced or getting enhanced and and Hopefully that will satisfy their suspicions Okay, at this point I would tell you an Open-stack joke But you'd have to build missing your part missing parts yourself to make it funny. I Think it's the only Open-stack joke in existence No cost proof of concept strategies So the obvious one dev stack I can start install it on Texts can the system ins can install them on their machines. I Give them some extra machines to install it on If you got some raspberry pies hanging around you can make a cluster out of them We grabbed spare desktops that were either waiting to be deployed or might be a little bit older and Started and the proof of concept started separating out all the components onto individual machines There's a lot of free trials that are out there rack space Moranis tri-stack they'll give you some resources storage space VCPUs and You can use that either to Build your help design your architecture or or for training in Lot of hardware vendors. They'll give you a couple of chassis Which is will be a larger set of of hardware that that would be available in the previous suggestions They'll give you a blank slate. You can actually practice installing the Open-stack on it and And testing the components out So go over some Some architectural I mean there's lots of papers out there and architectural design But there was some key points that I found Let a lot of talking to people that they they they missed out on or they didn't quite catch so When you're designing a file Production infrastructure you should spend more up front in hardware to To capture to for future capacity So if this is what we're using today, and this is what we think we're gonna be using in six months from from now Buy a little bit more Because then you won't have to be spending time adding the hardware in in later. I Cannot stress enough about high availability so instead of One storage node one compute node and one controller node We want a lot more You look at some of the design documents architectural Recommendations and you'll see it's a minimum of three controllers Whatever compute you have and you spread your storage out over as many nodes as you possibly can Best security is a stack-separated network where all the admin storage Networks are all separated from each other again. There's lots of documents out there that will explain and show how to set set this up, but it's Absolutely essential to to security you don't want one of the VMs Accessing one of your directly accessing one of your storage nodes and Very important in architectural design is your team so you want to Build the right team we divided them in a three layers where we have our core architecture architects system architects that were actually the brain trust for Open stack that Taken a lot of training spent a lot of time learning it and actually went through each level of the process and Does not help design and build and and test and test the systems so as you're doing that and if Your projects progressing well you can start Training some of your Linux sys amends They could have become architects down the road but you want to show these guys depending on the infrastructure you're coming from that how did How these system works how are they going to operate and how are they going to utilize?  You want to train them in some of the basics of Issues that can go on and how to fix and correct them and And we'll get into some other pieces of that in a bit Very important at each level of the Sandbox proof-of-concept pilot and production is to do testing At each level Don't wait to the end to do all your testing You will Find out you've made a lot of mistakes things don't work And you'll have to go step back on phases and explain the management why you're now going to take two more months to deploy this So there's some very good tools out there for testing operational use cases Tempest is one of the best rally is even better It actually uses tempest and has a UI and produces some better grass and some text output So you want to you know exercise all the API's that are possible on on the system and Try to break open stack We've not had a lot of success of that actually Now you're going to own the system the over the infrastructure is going to belong to you You're going to be supporting it now So one of the items that I've seen people miss out on is monitoring and alerting so A lot of these things in cloud providers happen in the background You'll get your alerts at your VM is down, but the underlying infrastructure. They're looking after well now you're looking after it So make sure that you find a tool that's going to monitor every single aspect Of of your open-stack infrastructure from the hardware up to neutron to the different components Of this is a Zenos by the way. You actually has a Zenpack. It does an awesome job already of Monitoring and alerting your system Wow, so alleviating risk Don't need to do this right from the beginning But as you go along you're going to realize what risks are involved some are obvious Like it you can own your own infrastructure now You're going to have to be worried about the power on on the bus on in the data center The networking into the data center you're gonna have to start worrying about now. So our risk register was about 22 pages long of items like this and Some of them a major some of the reminder, but they all still had to be Mitigated items such as Well, there's a who's gonna monitor who's gonna look after any of the security? Holes that are discovered in open stack or a component of open stack Who's going to be upgrading it? So and then what's involved in that process of upgrading? Well, do you have to bring the system down? Can you live migrate what's going on? So This should be identified you start this right from the sandbox, but it needs to be completed by the pilot To alleviate for design for high availability Don't even think about anything else if you cannot design for a high availability and in your components then don't even bother moving to open stack Because it's going to components are gonna fail hardware is gonna fail Somebody's gonna do something on the system. That's gonna bring something down So As far as the guideline goes have more than if it's possible to have more in one region that your Disaster recovery either as a live system or as a failover site Have more than well, obviously have more than one compute node. There's an awesome feature called live Migrate words of a If a compute node fails you can bring them up on a different compute node If you're gonna design it so you can have more than one compute node put them in zones have One zone per Tassie But no more than four zones and then place one controller on each one of the AZs data Durability if you One of the worst things is going to be you lose all your data all your volumes We went with cinder on sef and we went with a replication factor of three I mean there's three copies of All the data on the entire system that allowed us to do upgrades the systems If one got corrupted or one failed We had we had at least two other copies We used glance which is on on Swift as well And we also used a maids Swift multi-regional. So it would automatically copy it over to a different region Two regions for zones with a replication factor of four for a Swift Build a flat and fast ethernet fabric layer two There's The hardware is now becoming Less expensive and it's easier to get into a four-year or 10g equipment now and Bonded nicks off of some of the nodes and Low degree of over description. I put the sign up in The system in area no hacking at any time. So if you start hacking in the system and Changing out too much of the code or any of the code for that matter. You're gonna have some difficulty upgrading in the future I have a friend who works for a company and they're stuck I've been stuck on Folsom because they modified the open stack code so much that the features that they need they they built into it They absolutely need and they can't upgrade I told them if it's not a repository you cannot use it if you Want to add another feature or do something then you need to push the code up and have it accepted then you can start using it But otherwise There's a lot of architectural guidelines out there. Stay within the architectural guidelines whether it's either from the hardware vendor or Moranis or or Rackspace or whoever They're all widely documented. They've widely tested and they're all widely used. So just All the work's already done for you stay within them. You'll you'll be safe There's another hardware vendors out there that have open stack Deployments and I consider that a risk the whole opens part The whole advantage of open stack is getting away from vendor lock-in Having a vendor that insists that they're gonna you have to install their version of open stack Locks you in you've just locked in both the hardware and that open stack deployment and whether they Will add in features to it or you're basically back in the same situation you were when you first started out Partner with an agnostic support provider when I say agnostic I mean somebody is not going to push you towards a certain set of hardware or or a certain solution Yes, then stay within their architectural guidelines But somebody that's going to support you All the way help you install it help you to design it. It's going to Be a cost. Yes, but every dollar you spend is going to be well worth well worth it There are some proprietary open stack component solutions out there for different pieces either for management of a certain or deployment of a certain component Try to avoid them if not Then you know think think very very carefully again, that's going to lock you into their solution Set realistic expectations with open stack as you're moving through the pilots and the proof of concepts if you overstate or describe a Component or or a feature of open stack that Is not really there or doesn't really work as advertised You're going to have to you know after deployment What can you do? It's it's So I mentioned all those saw those other different features of of open stack that perhaps You don't have a use for today Well when you do your initial deployment don't deploy those they can be added in later So you want to reduce your complexity footprint? When you're doing your very first deployment first of all at your first deployment second of all You don't want to make it as complicated with with the other components you can add them in later and The learning curve is going to be steep So make sure that everybody understands in your team Understands exactly what's going on. They're not overstepping their their knowledge level if they are stop and identify the part that they might be a little weak on and Have them brought up to speed either themselves or through some other training and then and then continue on any questions Well, I didn't have that much I didn't have that heart of a time they Open stack is cool so It was I started off as a director very hands-on director so I started off with it myself and showing them on Putting a Dev stack on my laptop and showing them, you know, we're out for Lunge time or whatever and I'd show them exactly what the capabilities are are Tech guys like new technologies and this is very new And like I said, you can explain to them. It's very good for their career it's One of the worst things that that that Systeming can do is is stop what he's learning. So this is something new for them to learn Intellectual curiosity it it kind of just happened. So it's just really exposure And knowing that they have the support for it as well So the applications that we Everything revolves around a database, right? So the application that we moved first at one point We inherited our Infrastructure for the most part for the most part I did and then of course the guys that came after me did These systems were just Simply installed no documentation. We started building documentation for this as soon as we realized we want to move to open stack at some point we created runbooks and Decided on some technologies that we'd use to start the deployment Ansible For instance, so even if a new server was going into the current Infrastructure we would also in open stack in the lab write and build an equivalent So we had about 25 of our infrastructure that boom just moved over the rest we had to actually go through and like I said, we didn't build it we just supported at that point and and decipher exactly how applications were installed Some it was easy some it was very difficult The first ones we moved over Were the ones that were not dependent on the entire database infrastructure system. There were some supporting servers there were some servers that infrastructure used and The last part was the actual database applications and web servers and whatever How we moved it over we couldn't copy the images over how we moved it over Was rebuilding everything in open stack again, however, there's a vanish to that We actually got to rebuild everything document everything and build it correctly and more up to date Thank you very much


In this video we're gonna prove the second isomorphism theorem for groups. Analogous theorems can be stated for other algebraic categories like rings and modules, vector spaces and the like, but this video will focus on the second isomorphism theorem for groups. So let G be a group, this could be finite infinity group, doesn't matter. Let H be a subgroup of G and let N be a normal subgroup of G, okay? And so then the second isomorphism theorem under those assumption tells us that if you take the subgroup HN and you mod it out by N, this will be isomorphic to the group H modded out by H intersect N. So before we dive into the proof of this thing, let's make sure that this statement even makes sense because if we go about trying to prove nonsense, well, that's gonna be a big waste of time. Now note that by assumption we have that N is a normal subgroup of G, okay? So since N is a normal subgroup of G, this means that N contains all of the G conjugates of elements of N. So N here is closed under conjugation. In particular, N contains all of the H conjugates of elements from N. So let me elaborate a little bit more about that. So because N is normal in G, we know that for all say N inside of N and for all G inside of G, woops, we'll have that G and G inverse is inside of N. That's what I mean by it's closed under conjugation. But G, right, has H as a subset. It's a subgroup, but in particular, it's a subset. So if it's true that for all little G and G, it's closed under conjugation, I could restrict my set to something smaller, right? H will contain fewer elements than G. So if I restrict the set that I'm gonna conjugate by, then I'll still be closed under conjugation because we're a normal subgroup. So if N is closed under G conjugates, then N will also be closed under H conjugates. So that's an important observation to make right here. Also, well, I guess the conclusion we make from that is that notice N will be a normal subgroup of H times N. So we proved previously that HN is gonna be a subgroup of G. So first of all, HN is a group because H is a subgroup and N is a normal subgroup, okay? So it makes sense to talk about N being a subgroup of that. Now clearly, N, which is just equal to E times N, this is gonna be a subset of HN because H contains the identity. So the fact that N is inside of HN makes sense. As N is a group inside of another group, it's a subgroup, that's what that means. So the fact to say that N is a subgroup of HN follows from the fact that HN is a group. But the reason we can say it's a normal subgroup is this conversation up here that we're closed under conjugations of H. But in fact, I said it's closed under conjugations of H, we could also be like, oh, it's closed under conjugations of HN. If I take some element X, which is inside of HN, then you're gonna get these conjugations right there. So that's what I'm trying to say here, that is we restrict the group and it's still closed under conjugation in that regard. So summarizing what we're just saying there, we can say that N is a normal subgroup of HN. The reason this is significant is that this quotient actually makes sense. N has to be a normal subgroup of the numerator there, which it is, in fact, the case. So on the left-hand side, we have a legitimate statement. That's a legitimate quotient group. What about the other side? Since HN mod N is well-defined, can we say the same thing for H mod out H intersect N? Sure, we can. Well, from what we've also seen previously, we've seen that H intersect N clearly, if you take a set intersect another set, that's gonna be a subset of one of those things. So like H intersect N is a subset of N. But since the intersection of subgroups is itself a subgroup, we get that H intersect N will be a subgroup of N. And then by the same argument as we said before, H will be closed under H conjugates because that just comes from closure and multiplication in H. And as N is a normal subgroup, it'll be closed under H conjugates as well. So H intersect N will be closed under H conjugates. So in fact, H intersect N is a normal subgroup of H. We've proven this explicitly previously. So this tells us that in fact H mod H intersect N is also a well-defined quotient group. So that whole discussion is just to make sense that these two groups are well-defined quotient groups. And it turns out that the same quotient group up to isomorphism. All right, so how are we gonna prove this? So it turns out when one wants to prove that two sub, or two groups are isomorphic, there's essentially two strategies to do that. I may be saying there's three strategies, one proves this. So the first strategy is to actually design an explicit function, a specific isomorphism between the two. It's like, here's the formula for the isomorphism that shows the two groups are isomorphic. That's a nice strategy. Another one that's a little bit more difficult and this is the one I'm adding to my list of two that became three, is if you do have some type of classification theorem, kind of like the fundamental theorem of finite abelian groups, there could be like a certain sufficient collection of invariance that when they agree, they determine the group. And so you could determine that two groups are isomorphic without an explicit isomorphism because they get some classification theorem like the fundamental theorem of finite abelian groups there. That's an awesome nice strategy if you have such a classification theorem. Those were sometimes hard to come by. The third strategy, which actually I would think in practice is one of the most common ways to prove two groups is the isomorphic is use some type of universal mapping property. That is a condition such as the first isomorphism theorem that guarantees the existence of an isomorphism. In fact, perhaps like a unique isomorphism like in the first isomorphism theorem. If we can guarantee the existence of a isomorphism theorem of isomorphism then the two groups have to be isomorphic. The main difference here is that the universal mapping property might not make explicit what that is. The isomorphism is sometimes implicit. So what I'm trying to say in a nutshell is you don't wanna underestimate how powerful the first isomorphism theorem is. The second isomorphism theorem we're gonna prove using the first isomorphism theorem. And this is a very important strategy for a budding algebraist. How can you prove that two groups are isomorphic using the first isomorphism theorem? So remember what that says. The first isomorphism theorem tells us that a group if you mod out by the kernel of a homomorphism will be isomorphic to the image of that isomorphism of that homomorphism. So that's what we're gonna do right here. So what we're gonna do is we're gonna establish a homomorphism from H in to H mod K where K here is gonna be H intersected. So really what I'm saying is phi is gonna be a map from H in to H mod out H intersect in. Like so. So basically what I'm doing is I have two factor groups I wanna show are isomorphic to each other. So I'm gonna take the numerator of one of the factor groups and show that the other factor group is a homomorphic image of that numerator. Then we're gonna argue that the denominator of the first factor group its kernel is in fact equal to N. Then we conclude by the first isomorphism theorem what we want. So we're gonna try a homomorphic way of proving these two groups are isomorphic. So what's the map H in? Moving over to H mod K where again K is H intersect in here. Well essentially there's only one thing we really can do. If you take a typical element of H in and it'll look like little H times little N, I'm gonna map this over to the coset HK or again K is H intersect in. Okay, so let's first make clear that this is a well-defined map. So suppose we have two different factorizations of the same element of HN. There could be such an element, right? So let's imagine we have HN which is equal to H prime N prime or H and H prime are inside of H and N and N prime are inside of N. Well, if you take this equation here HN this is equal to H prime N prime. Let's multiply on the left side H prime inverse. H prime inverse here, H prime inverse in which we see that the H primes will cancel on the right-hand side. But let's also take N inverse of both sides so that the N's cancel on the left-hand side. This will show that H prime inverse H is equal to N, excuse me N prime N inverse. Well, the term on the left, this is inside of H because it's a product of two things in H, H is a subgroup. The terms on the right, this is gonna be something inside of N, N is also a subgroup, a normal subgroup mind you, but it's a subgroup. So product of things in N are gonna be an N. So we have something that's in H and in N. So this element, let's call it X just to give it a name. This element actually belongs to H intersect N AKA it's inside of K. So with that in mind, let's consider the image of H prime N prime. Well, by the definition of our map right here, H prime N prime will map to H prime K. But since X belongs to K, right? This right here is just K. Since X belongs to K, you could replace H prime K with H prime XK, right? X represents K just the same way the identity does. And then re-associating this thing we get H prime X, K, right? Essentially, if you have an element inside the group, the group can barf it out without changing the coset whatsoever. So you're gonna get H prime X, which look at X right here, right? X is equal to H prime inverse H. So if I take H prime times X here, you're gonna get H prime inverse times H. Then we're gonna see that H and its inverse will cancel out, H prime and its inverse will cancel out, giving us just an H. So we get HK, which is the image of phi of HN. So this tells us that the map phi is in fact well-defined. It doesn't matter on which factorization of the element you use. So now let's prove it's homomorphic. So if we take the product of two typical elements of H, the capital H, capital N, I'm gonna use HN and H prime N prime. So in this situation, we're not assuming that HN and H prime N prime are equal to each other. There's two arbitrary elements of our group. Well, because we are in the group, this product is gonna equal something in HN.  Because N is a normal subgroup, this is allowed, H prime will move to the left, but then the N might switch to a different element of N, capital N. Let's call that new element maybe little N, double prime, like so, for which then here we have something in H, we have something in N, in which case phi will then send it to the coset represented by that element of H, H times H prime, which as this is a coset, it's in fact, this is a coset where K is a normal subgroup of H intersect, excuse me, K is a normal subgroup of H. We get that H prime K is equal to HK times H prime K. You can factor it inside of the quotient group. Now, HK is just the image of phi of HN and H prime K is the image of H prime N prime. So we see in fact that we have a homomorphic map, all right? Then the next thing we need to do, now that we have a homomorphic map to show that it's surjective, this is kind of an obvious statement, right? Because remember phi is supposed to go from HN over to H mod K. So what we're gonna do is notice that you have the element H, like H is a subgroup of HN, right?  And so this right here is going to map H, HG, which is just H of course, it'll map down to HK, which if H is an arbitrary element of capital H, then HK will be an arbitrary element of H mod K. So we have that. It's gonna be surjective again, that's a triviality. So then the other thing we had to show, remember the other thing we wanted to show was that the kernel of this map is equal to N. If we can do that, we can invoke the first isomorphism theorem. So let's consider what elements HN will map to K, which is the identity element of H mod K. Well, if HN maps to K, I mean it should be mapping to HK, this would imply that H is inside of K, which remember K is H intersect N. Clearly H is inside of H, but this also tells us that H, little H has to be inside of N, all right? And so this is what then implies for us that the kernel of phi is gonna equal N. Those elements, those elements that live inside, and I mean I should also mention of course that, so if H is inside of K, right? Clearly N is inside of N, like so, this would imply that HN is gonna be inside of N, like we're trying to say here, which is a subgroup of HN. So this is where we get this inference that the kernel is going to be all of N. Again, as we observe right here, if you said N, N will just map to the K as well. And so by the first isomorphism theorem, we have this map phi, which goes from HN to H mod K, for which the kernel, the kernel of phi is equal to N. So the first isomorphism theorem tells us that HN, the kernel of phi, is gonna be isomorphic to the image of phi, which fill in the details there, we get HN, the kernel we've established was N, that's the left-hand side of what the second isomorphism says. And since the map is surjective, since the map is surjective, the image of the homomorphism is the co-domain, which we saw earlier was H mod K, which K was just a substitute. This is H mod H intersect N, like so thus proving the theorem. Let's look at a quick example of this real quick. I said quick twice there. So let's take as a group, the symmetric group on four letters, S4. Let's take the Klein-4 group, which is a normal subgroup in there. And so by the Klein-4 group, of course we mean you have the identity and you have the two two cycles. Well, let's see how those elements right here. That's what we mean by the Klein-4 group. And then let's take the subgroup H to be the subgroup generated by one, two, three and one, two. So you can really think of like, this is S3, this is the symmetric group on three letters sitting inside of the symmetric group of four letters. So H is a subgroup of G and as a normal subgroup. So notice those things have been established, right? H is a subgroup of G because that's really just saying that S3 can be viewed as a subgroup of S4. And we also have the Klein-4 group as a normal subgroup in S4. So we have the assumptions necessary for the second isomorphism theorem. Let's see what it tells us. Let's consider the set H in for a moment. We've learned previously that the set H in, the order of this subgroup is gonna be the order of H times the order of N divided by the intersection H intersect in right there. Well, H is a subgroup of order six, it's S3. We get this one right here and which is the Klein-4 group, it's order four. And then what elements are common to both the Klein-4 group and S3? That turns out just to be one. That is the identity, it's the only thing common. So you get six times four times one are divided by one. That gives you a subgroup of order 24, right? Because the intersection is trivial. Well, as S4 itself is a group of order 24, four factorial, the only subgroup of S4 that says order 24 is S4 itself. So this tells us that HN is equal to S4. So let's consider then these quotients. If I take S4 mod V4, so the symmetric group mod out the Klein-4 group. Well, S4 is the same thing as HN. The Klein-4 group is N. By the second isomorphism theorem, this will be isomorphic to H to mod H intersect N which H was the symmetric group S3 and H intersect N was trivial, right? So if you take a group and mod out the trivial subgroup, you're gonna get S3. Well, you'll get the original group back, which is S3. So this tells us when we put these things together that the symmetric group on four letters mod out the Klein-4 group is in fact isomorphic to S3.