All right. What's up guys? This is Alex from Xtrades. Back to you with another weekly trade ideas list. Hope everybody had a wonderful trade week last week. We had an amazing list last week actually. All the setups were pretty nice. I think there was just one that really didn't get going. That was probably Adobe Puts, but the KO calls, AT&T calls and the long-term position that I mentioned I was going to be starting did really well closed AT&T calls for 85% in our Xtrades Discord, and I'm holding shares for the long term intending on collecting the dividend. There was also MU Micron. They did pretty good. They had a breakout. It was not able to reject the downtrend, so we're looking at calls or puts on both of those. The puts were not able to confirm a 30-minute close under the level we're looking at, so it ended up breaking out of the downtrend and actually gave us setup for calls. So that did pretty good as well. And then lastly, SMH, the semiconductor ETF, finally broke down the level on Friday, or maybe it was Thursday or Thursday, and Friday both kind of had a red day, but it was able to break under that level and the puts worked out pretty good. Just took a couple days and there was a little bit of upsides. You just had to wait for that level to break, but hopefully this week we'll have another good list and we'll have some good setups just like last week. I had a really good week personally. Probably one of my best in a couple months, so that was really nice to have a steady equity curve this week and just overall be able to give you guys quality setup. So this week, hopefully it goes well. We do have the FOMC meeting this week, but we'll go over an economic calendar here. So this week, Monday, July 24th, we do have the S&P Flash US Manufacturing PMIs and also the S&P Flash US Services PMIs. So this could definitely move the market. It's a pretty good data set. Just depends on how extreme the reading is. Tuesday, July 25th, we have the S&P Case Schiller Home Price Index and also Consumer Confidence. I wouldn't say that this one's really going to do much, but the Consumer Confidence can definitely move the market. Wednesday, the most important day of the week, we do have new home sales, also the FOMC decision on interest rate policy and also Fed Chair Jerome Powell also has a press conference that's at 230 as usual. So this is definitely going to be really volatile. I believe there's a 99% chance they're going to raise 25 basis points. According to the Fed Futures and usually Jerome Powell does not shy away from the Fed Futures. So we could be seeing a little rate hike coming up. But honestly, with the last CPI print, I feel like they could pause and they'd probably be good because policy does have a lag effect and it takes a while to catch up. And I believe the CPI is already down to 3%. So their goal is 2%. And I mean, they're really close. So honestly, I feel like they can pause probably and inflation would keep coming down. According to their data is coming down so fast. So, but you never know, we'll have to see what they're thinking and we'll get all of that on Wednesday. So it's probably pretty important not to swing any big positions through this, honestly, just because it's so volatile and random. If you were to, you'd probably just want to stick to your long term positions. Obviously, you're probably not going to be selling those for a really long time, such as myself, I'm not going to be selling my long term positions unless there is a very good reason like cutting a dividend or like a real economic disaster. So, and honestly, even through that, I probably wouldn't sell because the market usually bounces back. So, you know, just be smart, make sure you have time on contracts if you're playing options. And yeah, just be careful swinging through stuff like this. It's crazy. The implied volatility gets high. And overall, it's just very random. So you just never know how it's going to go. I wish there was a way to project where it would go during an FOMC rating, but it's like literally impossible. You know, you could take an educated guess. And usually the market tries to be, you know, two or three steps ahead and always pricing in the market to current conditions. So, you know, some people will argue everything's already priced in. So the market's already ahead of the point 25 basis point hike. You know, you just got to be careful with that. So and Thursday, July 27th, our usual initial job was claims, we have durable goods orders. Also the GDP report, it looks like we have advanced retail inventories and also advanced wholesale inventories. And then pending home sales. So this is a pretty good data stack day. I would say the ones I just read off or probably ones that can definitely move the market, it just depends. And I feel like there's going to be a lot for the market to digest here. So, you know, we could see some movement. And then also, it's the day after the FOMC. So it's likely going to be a digestion day anyways for the FOMC decision. So FOMC decision can probably just, you know, overweigh all of this right here. And people are just going to be trying to price in what the Fed said during the press conference. And Friday, July 28th, we do have personal income, personal spending, and the PCE index as well, core PCE, PCE year over year, and core PCE year over year. And this is the Fed's preferred inflation gauge. They prefer this over the CPI. If you watched my videos before, I've pretty much repeated that a couple of different times. And they've made it known that they prefer the PCE over the CPI. So this is a pretty important reading. It does come after the CPI. So it kind of like loses some magic because we already got the CPI print. But it does move the market a little bit, I would say anywhere from, you know, half a percent to 1%, it can move the spy. But like I said, it just depends on how extreme the reading is. And then we do have another consumer sentiment on Friday, which is mid session. So this comes after the bell rings at the open and can cause some mid session volatility. So that's for the data, pretty stacked week, I would say a lot of these have a potential to move the market. But most importantly, Wednesday, July 26th, FOMC decision, and also the press conference. So it's going to be a big day. And then we do have another edition today, because we are going into a very earnings stacked week. So this is just adding more insanity to this week. So this week can be extremely volatile, along with the Nasdaq rebalancing some of its weighting, and some of the biggest big tech stocks that they have large weight into on the ETFs, they are starting to rebalance that I'm not sure what stocks they're doing, but they are taking down some of the big ones, like taking some of the weight off. So they are lowering the weight percent on like Microsoft, I think like Tesla, and maybe a couple others, I'll probably have to go back and look, but they are doing a rebalance on some of the big tech stocks. So they are probably going to be selling some of that. And that can cause volatility, along with it being the week after the monthly options expiration. Usually there's some weakness after the after that anyways. So I would say that, you know, this rebounds with the Nasdaq and the ETFs can just, you know, cause some additional volatility along with the OPEX weakness, because people are rebalancing their portfolios, trillions of dollars just expired. So kind of makes the week after a little bit weak, because people are rebalancing their portfolio after such a big expiration. But for earnings this week, you can see we got Domino's Pizza, we have Cleveland Cliffs, this is a nice steel ticker to trade. AG&C, it's a pretty nice REIT. I've actually invested in this company before, a very long time ago. We have Verizon, 3M, GE, Gemro Electric, Spotify, Raytheon Technologies, Texas Instruments, Teladoc, this is a high growth company. Kathy Wood really likes this one. We got Visa, we got Snap, Alphabet Google, and Microsoft. So Tuesday is going to be pretty huge. And then AT&T. So this should be interesting after AT&T is just been getting slammed to finally have that dead cat bounce. We'll have to see how this goes. AT&T does have a lot of debt and with high interest rates, that does make it a little bit riskier. People want to see that, you know, they're still holding on to their dividend, they're still going to pay shareholders. And yeah, hopefully we get some good news on that. We have Boeing, we have Coca-Cola. We had this in the list last week, had a really nice pre-earnings run up, calls paid well. Tilbury, if you like pot stocks, we got eBay, Quantum Skate, Chipotle, Meta. So we got Google Meta and Microsoft. So that's some of the big seven, some of the biggest tech stocks. We also have RCL, so cruise ships. We got McDonald's, this is a really nice value names, one is just great and recession proof. We have LUV, so some of the airlines, we got Mastercard, Crocs, we got Mphase Energy, we got Ford, Intel, Roku, T-Mobile. Then Friday, it looks like we're getting some of the big energy names. So we got ExxonMobile, we got Chevron. So you can just see, it's just very stacked, it's almost like all of America is reporting next week. And then we have, you know, the FOMC meeting too, so it's going to be very stacked. So just stay in our pivot and make sure we're not, you know, oversizing and taking too much dumb risk until, you know, maybe some of it blows over and there's a little bit more of a clear path for us to follow. All right, and now that we've gone over all of that, we went over the economic calendar and some of the big earnings, which is just very stacked. We're going to get into some of the setups we have. I only have three for this week. Last week we had five, but I could really only find three that I actually truly liked in a lot of, I did find some others, but they're reporting earnings. So your technical analysis going into earnings can just be completely blown out of the water and it could not matter. So it's just good not to trade options and swing into earnings unless you, you know, you got money you're willing to lose, or you just buy really further out expiration and you deal with the volatility. So the first one here, we're looking at Nike.  I would like to see a little bit more confirmation though, maybe one more bar closing maybe like above 110 or something showing that it's finally breaking out. The price target is pretty obvious. There's really only a supply up here, you know, around 114 or so, maybe like 115. And you can see that the slow stochastic also still crossed that positive no negative crossover yet. So momentum is holding positive, but I would like to see it definitely close outside of the downtrend line, maybe one more time or so. And this could be a great breakout trade to the upside. So I'm going to be looking at calls on Nike. As long as we get a good signal, want to see a one day bar closing outside of this, just one more, maybe if you want to be a little bit more sure, you wait for it to get over this one 1034 short term resistance, and you can even right click, hit add alert, and we can just name it breakout. So once it gets over that one 1034, that's a pretty good signal that it's probably going to start feeling this little settling bounce area, and just head straight into supply. So pretty good setup. But like I said, it's kind of close to the downtrend line still. So I want to see a little bit of a better signal that it's breaking out of this downtrend line. So Nike here looking at calls. All right, next, we're going into Netflix. So they actually reported earnings last week, huge gap down, closed down almost 9% after their earnings report. So the subscribers are actually really good. So they added a lots of new subscribers. But I believe that their revenue really wasn't all that and people kind of, you know, raise their eyebrow with that how you get all these new subscribers, but your revenue was kind of mid. So that could be why it's so rough, you know, could just be people taking profits as well.  But me when I'm looking at longs like to see discounts. So now that I see that it's sold off a little bit, it's pulling into this nice demand zone. This is a previous looks like a pretty much like a rally base rally demand zone and bounced off here prior. And now it's back to it again. So I'm going to be looking for a little call scopes on this. I don't think I'm going to open a swing trade on it, just because I don't know, I don't really like the structure here.  So I need to see it maybe entering the gap. And then I'll be willing to take a couple days swing trade to fill the gap. But buying down here while I could, you know, run right into 448 and make a shoulder, I feel like I wouldn't really like the risk to reward on that. So I probably won't swing this, but I do like this for day trades. So as long as this little demand zone's holding up, you know, we can trade off of it. And I feel like it's a pretty good demand zone. I mean, it's already had one good bounce from it. So let's see if it can hold up here. I'm going to be looking at calls on this, obviously risk off and it just goes totally void. Probably if it goes under 410, maybe even this little 41150, which is this little Wicklow right here, the demand zone low is precisely 41476. So you could use that as well. But I feel like this Wicklow would be pretty good because if it bounced to this Wicklow, it could hold up right there. So we'll just call it 41150 as the risk off area. Probably wouldn't be looking at any long trades if it broke that, unless I want to close back over it, reclaim it, you know, give a nice little bullish candle, something showing us that the support or the demand zone is still holding the lows.  It's usually how it goes, but like I said, it just depends. So we'll have to see how it opens on Monday. When it closes red like this, there's a good chance you'll see that continuation on Monday. So you might see a dip into Monday, and maybe people will try to buy back up into Tuesday, but it just depends. So we'll see how that goes. Just keep an eye on the demand zone, and maybe be willing to scoop up some calls for a scalp or some type of day trade. And then you can be flat by end of day, and then risk off under 41150. So Netflix here looking at calls. All right, and we got another big tech name here. So we're going to be looking at Nvidia. The reason for that is because it finally dipped. Now that it finally dipped, it's actually pulling into this 43990 back test area, you can see right here. So this is the 43990 back test. It's a previous resistance. And you can see it's just pulling in for a classic, you know, breakout retest. And it could try to hold up here. Another thing you have is a test one, you got a test two, test three, test four. It looks like the trend line is coming up for a test five. So there's a good chance that this could still hold up. I know this drop is kind of sharp. So it's a little bit scary buying when it's, you know, so high up and you have this sharp of a drop. But that's why you make sure that the 43990 and the trend line is holding first before you enter. Another thing I did here while I was charting this out, you can see a little clock down here that means I right clicked it, I added an alert and I named it breakdown just in case this doesn't end up holding up. Look, let's say for some reason, we just got down super heavy on Monday, obviously that would set off our trigger and, you know, that being the trend line broke. So we could probably start looking at puts at that point, but right now it's still over the trend line. So naturally, at first, I'm going to be looking for it to hold up 43990 plus the up trend line. Only thing I don't like is that a closed week on Friday could just be from OPEX. It could be, you know, just people taking profits. But like I said, you know, when you have a red Friday like this, a lot of times Monday, it will continue. And then people will buy it up going into Tuesday. So it's the down Friday, down Monday, buy Tuesday. It's kind of a, it's a pretty common pattern we see a lot, especially in the spy and QQQ and just overall really with the market in general. So this can go two ways. Like I said, as long as we're holding up trend line and back test level, I'm going to be looking at calls on this. But at the same time, I also did set that alert just in case it wants to break down and we can switch. So trading is just all about, you know, willing to adjust and markets change in seconds. So I mean, the plan could change, you know, we'll have to see how Monday opens. We'll see if it's holding up 43990 plus the trend line that could give a signal to bounce back up short term, look at a day trade on calls. I probably wouldn't swing up here just because it's, I mean, it's so high up. We're at 440, we're at 443 a share price to earnings ratios outrageous. I'm guessing it's 200 plus the S&P average is anywhere from 15 to 20 price to earnings ratio. So that just, you know, this is a high growth stock. It's very quote unquote overvalued. So you have to be careful. I'm guessing insiders are probably want to start taking profits on this soon as well. It's had a great run up. So it'd be silly for people not to, but we do have this uptrend line in place. So we'll see if this can hold up here. Maybe we could look at some calls. If not, we'll have some insurance, the trend line breaks, you see the little clock, we set the alert, we look at puts to the downside. So Nvidia here, I'm willing to do either, but focusing on calls going to Monday as long as this is holding. All right, and now we'll go ahead and get into the indexes. First, we'll go ahead and look at spy here. So last week, we were focused on this rally based drop supply zone right here all the way from 2022. And we were focused on the 78.6. So I really didn't feel like it would go too much higher because the supplies in the way, and also you have the 78.6, it did get briefly over that. But now you can see the close right, right around the 78.6. You have a nice little wick on this one week bar. So it did react to the supply. And it looks like the 78.6 could still be a factor, but it would have to get back under it. Obviously, this could be a, you know, 451 or so needs to get under that. But it's reacting to the supply. Another thing we have right here, we have two little base candles for demand. So if spy does pull into this little 450 area, this is a one day demand zone. The big red you see over here is a one week zone. So this one day zone is a little bit more recent. So we definitely want to market just because it's recent and there could be fresh liquidity waiting to, you know, buy around this area. So if it does pull into the 450 20s, you got about $2 still does that, we could probably look at, you know, call scalps to the upside. I probably wouldn't open up a new swing trade here. I probably want to wait for it to get much lower to buy a swing trade on spy for calls, maybe like 443 or so, or maybe even down here. But up here, I just don't feel great about it just because it's still in the one week supply. But for short term, we can focus on this one day demand zone. So if it pulls into this, we can look at calls off that, maybe trade like $1 or $2 to the upside and, you know, call it a day. Even if you have to trade $1 or $2 on spy, you can, you know, catch 50 cents on some zero day contracts and they'll pay, you know, 20, 30%. So just all about perspective. Like I said, still inside this massive one week supply zone. So definitely not opening any swing trades up here yet. And go ahead and add the moving averages. I would feel better if it pulled into, you know, the daily nine EMA, maybe even the daily 21 EMA. And I would be willing to buy calls around there to hold for a couple of days. But up here, just not for me, especially with the supply in the way. And we're pretty overextended. So we'll just focus on this demand zone. I feel like it could dip a little bit lower. It'll be about, you know, 450, 450 to $450 and 50 cents. So it'll probably be the area where we're trying to, you know, curl up. And that kind of meets with the daily nine EMA too. So whenever it does pull into this, or if it does pull into this, pull up your one day chart, see if it's tapping the one day nine EMA and your demand. And it could be a good area to enter, you know, a nice little 15 minute scalp or something to the upside. So I can't really put any lower than that. Just because we have this demand zone here, it would have to get under the demand zone, probably about 44850. If I can get under 44850, I could put it to the next base right here. But that's about it. I can't really project it any lower until I see these get wiped out.  But it looks like the range could be tight with the FOMC, obviously it'll widen up. But for Monday and Tuesday, I'm guessing that it probably won't breach this, but it just depends. We'll have to see. And then after Wednesday, I mean, anything's up for grabs. The market's going to get probably going to get really wide, either to the upside or downside. So, you know, these could flush right through. I probably won't hold any or I probably won't open any new swing trades, you know, going into FOMC, but we'll see. So just focus on this demand zone. If it comes down to 450-52, keep an eye out for a bounce just in case. And then obviously, just be careful with 450-52, because that's where we're, you know, the one week supply zone starts. All right. And next, we're going into QQQ. So last week, we were focused on this one week supply right here. And you can see exactly why. I mean, it finally started reacting to it a little bit. We actually had a pretty big dip, I think Thursday and Friday after Netflix and Tesla earnings, it brought down the NASDAQ pretty well. But I was pretty much skeptical in this area. I definitely said I wasn't going to be looking at calls inside of here. I think I did scalp some QQQ calls last week for a day trade, but obviously I only hold those, you know, 10, 15 minutes or something. So I think it worked out pretty good. And I think I did some puts as well. So I had a pretty good week last week trading the QQQ as well, along with some other stuff. As long as the camera, the pivots look good, I'm willing to still buy the dip at S3 or trade at R4 breakout, even if it's inside supply. So it just depends on intraday sentiment, how the VIX is doing, how the dollar is doing, etc. Same thing as last week. So we did close a little bit lower than our last one week bar that we were looking at. It didn't take out the week's week's lows or anything. So I don't think that's, you know, like a huge signal, but it's a start. So it did start to react to the supply. It's about the only signal we have right now in the one week is this little potential reversal candle on the one week. Looks like you have the slow stochastic starting to cross over a little bit. You can see the purple low line going over the orange. That means it's starting to cross to the downside. So it could be a little sign of momentum slowing, but we'll go down to the one day right now. It's easier to kind of project short-term moves or look for, you know, stuff for day trading. The one week's more for, you know, swing trading and more medium and long-term projections and ideas. So it looks like QQQ does have this previous resistance. It's going to be at 372.85. This is definitely a big level of focus. So definitely keep an eye on this because it could just pull back into this. And I'll probably try to curl up about there or at least try to bounce off of it. So even if it, you know, ends up flushing through, there's a good chance that it probably will try to try to hold up here at least as a classic, you know, break, break out, retest and try to bounce about there. The only thing you do have looming over you, like I said, is the one week supply. So we're conscious of that going into last week when I dropped the last video. For a good reason too, because look, I mean, it got almost all the way up to the supply high. So I had a good start.  So you just have to be really conscious of these one week supply zones, even if, you know, maybe you end up wrong and it just blasts all the way through. It doesn't matter.  And that would have done that for you right here for last week. So just keep an eye on the 372.85. Pulls into that. You could try to look for a call scalp of that maybe. And then obviously, if it breaks under the 372.85, that probably take you down to demand right here. That's going to be pretty low, you know, around, you know, the 360s. 362 or so, maybe even this little week low at 363.40s. And then it looks like there's some small gaps here as well. You can kind of see them. They're pretty small. It's nothing crazy. But we do want to keep an eye on those. We're going to even mark them. I think I just removed them earlier because they make the one week chart look ugly. So there's one gap right there. And there's a little gap right here as well. So it's just a little bit small. So it looks like it could fail down a little bit. And then maybe I'll try to, you know, bounce off the 372.85. Hold up there short term, you know, before trying to break it at least. So just keep the 372.85 as your main level of focus. And then obviously, you know, be conscious of the gaps your one week supply, which we're still inside of briefly. And then you do have a major demand zone down here from where we bounced prior. But most importantly, just the 372.85, look for a bounce on that maybe if it pulls down a little bit. And it could be a good area to buy the dip. So we'll keep an eye on that. We can even set an alert there just in case. And we can just call it support. Even though it's not a support yet, it could be a support. So we'll just call it support. Hit create. All right. And that's for QQQQ. Just watch the 372.85 and be aware of the gaps and supply zone. All right. Next, we're going into IWM, which is finally just had a massive run. I was actually able to break out the 189.24. So it briefly closed over 192 on Friday. And then I was looking for it to actually pull back into this previous resistance. It didn't even get all the way down there. It actually just held up. But it made it higher high right here. So this is actually a new rally based rally demand zone. And then if you zoom out right here, this is actually a rally based drop supply zone. And this is exactly why you see these little resistance areas right here. So you got two red candles off of this rally based drop supply. But we do have a new fresh demand zone. So if I can pull into this, I would look for IWM to try to hold up about there. So it looks like the spot QQQ and IWM are all very close to key levels. They just need to dip a little bit lower. And then you can, you know, maybe look at trying to buy the dip, at least for scalps or something. You don't have to enter swing trades or anything. Because I know the market's a little bit extended. You may not think this is the best risk to reward. So maybe just keep it short term. But you will have to wait for it to dip a little bit on all three. So spy was just a little bit above a level. QQQ was just a little bit above a one day level. And then IWM is just a little bit above this one day demand. So you might want to wait for it to dip just a little bit. And you're going to look to buy the dip around there. But it's a nice rally based rally zone. So you got a really nice rally here. It makes the base higher highs. So confirmed after this little candle right here. And then with demand zones, you just wait for them to return. And you try to buy about there. So it's kind of the same plan as last week. Except there's a demand zone instead of previous resistance. And last week, we didn't even get into the previous resistance. So it didn't hit, you know, even really that close for me to want to buy last week, unfortunately. So I wasn't able to trade it. And it really didn't follow the plan. But I mean, it's not always going to do that. So you're not always going to get your perfect resistance as a back test. So you might have to, you know, look for the general area, pay attention to other things as well. But like I said, all three indexes by QQQ and IWM just briefly over some major levels, some key levels at least for the short term that you can try to buy the dip at. But I would just wait for them to get down there.  And obviously just shortly above that, if it were to bounce off this, it does go straight into supply at about 196, upper 195s. So you want to be conscious of that. So that's for the IWM. Just wait for it to dip a little bit maybe and you can look at, you know, bounces off this demand zone. And next, we're going into the VIX, which pretty much just closed a little bit higher when we closed last Friday. It was not able to get over this 2021 low at 1410. It wasn't able to really do anything. If I add some moving averages here, it tried to briefly get over the 21 EMA. So this is that yellow line. You can see it popped over it, got up to 1420 or so, and then closed back on there. So we do want it to get over the 21 EMA to start seeing that short-term signal for it to start heading back up to 1553. So I would just keep an eye on that, wait for a moving average signal, and then, you know, maybe you could look at some VIX calls or, you know, look at spy puts. If it can close over the, you know, daily 21, because it's been respecting the daily 21 pretty well. You got a rejection of the general area here. You got another rejection right here. And then look, once it got over the 21 here, major pop, even got some volatility these two days when it got over. So the 21 EMA is a good area to keep an eye on on the one-day timeframe. So just keep an eye on that. That's about all I got for it this week. I mean, it's still holding over 1273. So it's still over, you know, the recent lows, which means it could, you know, just base out down here and try to pop. You can see the slow stochastic has actually crossed positive. So that's interesting. It's probably from these two green days we had. And then we just had a red day on Friday for the VIX. But like I said, I just wait for it to get over the daily 21 EMA, but you want to see a one-day close over that before, you know, just blindly following a signal. So we want to see a close over the daily 21 EMA and that could take you up to 1553, maybe even this daily 50 EMA. So just use the one-day EMAs. They're working pretty good for the VIX, at least for signals and short-term signals. Right now it's rejecting. So there's a good chance, you know, eventually over time, it's probably just going to head back into the lows as long as it's trending under the 9 and 21 EMA combo. So keep an eye on that. Keep an eye on the slow stochastic, which is positive right now. So that could be a little early warning sign. Otherwise, there's really no signals here. Just holding supports, trending under the EMAs. Need to see a little bit more from it. All right. And lastly, we're going into the DXY. So last week we were actually focused on a one-week demand zone on this. I felt that that was the maximum low it would go to. It actually didn't even get down there. So this is the Friday candle. We only dipped to the Friday low Tuesday, it looks like. So it was right at, you know, 99.59 or so. So the maximum I had it down to if it dipped was this demand zone low. And I would set it to look for a bounce there if it got down there. At the same time, we were looking at the one-day chart and I said it looked very oversold. So there's a, you know, a good chance that you would see a dead cab bounce as well as the slow stochastic had an early cross signal. And look what it did. It just straight ripped back up. And that could be due to the earnings, especially on Thursday, which the market just sold off pretty heavy, at least for the NASDAQ. And then I'm guessing that brought the dollar up as well. But now that's actually reclaimed over the 182, which is basically the most important support of the year. I mean, it was pretty much our structure hold. And it broke it very briefly and now back over it. So as long as this 182 area is holding up its support, I feel like it could bounce back up and, you know, head into 102 flat. If it goes back under, obviously, that takes you to the most recent low is right here. And this is all just a straight buy-in balance. So if it does go back under 182 and then 100 flat, it's probably going to take you back down to 99.57 at these lows. So it's all dependent on the FOMC meeting. Currencies are very, very sensitive to the Federal Reserve. So we'll see how that goes. But looking pretty good for a balance. I mean, it's over the 182 now. It closed over it with a one-week bar. And as long as it stays over that, there's a good chance the dollar's going to bounce back up. So keep an eye on that. Stay very frugal with that because if the dollar comes back up, after breaking this, this break under the 182, it could have been a great signal for the market to go higher. But it was very brief and now it's right back over. So you got to be careful with that. If the dollar goes up high and very fast, people get spooked and they start selling stocks. Not all the time. But there's times, you know, where the market doesn't care about the dollar, but eventually it does come back out of the woodworks. So you got to be careful with that. So just watch 182 as long as it's holding over that. I feel like the dollar could bounce back up. So that's the video, guys. Make sure you like, comment, and subscribe to our X-Trade YouTube channel. I love you guys. I'm going to go get this chopped up, edited and sent out. And I'm out.


From around the globe, it's theCUBE with coverage of KubeCon and CloudNativeCon North America 2020 virtual brought to you by Red Hat, the Cloud Native Computing Foundation and Ecosystem Partners. Hello everyone and welcome back to theCUBE's ongoing coverage of KubeCon North America. Joe Fernandez is here, he's with Stephanie Charras and Joe's the VP and GM for Core Cloud Platforms at Red Hat and Stephanie is the SVP and GM of the Red Hat Enterprise Linux BU, two great friends of theCUBE. Awesome seeing you guys, how are you doing? Good, it's great to be here, Dave. Yeah, thanks for the opportunity. Hey, so we all talked, you know, recently, Ansible Fest seems like a while ago, but we talked about what's new at Red Hat, really coming at it from an automation perspective, but I'm wondering if we could take a view from OpenShift and what's new from the standpoint of you're really focused on helping customers, you know, change their operations and operationalize. And Stephanie, maybe you could start and then, you know, Joe, you can bring in some added color. No, that's great. And I think, you know, one of the things we try and do at Red Hat clearly, building off of open source, we have been focused on this open hybrid cloud strategy for, you know, really years now. The beauty of it is that hybrid cloud and open hybrid cloud continues to evolve, right? With bringing in things like speed and stability and scale and now adding in other footprints like managed services as well as Edge and pulling that all together across the whole Red Hat portfolio from the platforms, right? Certainly with Linux and RHEL, into OpenShift and the platform with OpenShift and then adding automation, which certainly you need for scale, but it's continues to evolve as the definition of open hybrid cloud evolves. Great, so thank you, Stephanie and Joe. You guys got hard news here that you can maybe talk about the 4.6. Yeah, so OpenShift is our enterprise Kubernetes platform with this announcement, we announced the release of OpenShift 4.6. So we're doing releases every quarter, tracking the upstream Kubernetes release cycle. So this brings Kubernetes 1.19, which itself brings a number of new innovations, some specific things to call out. We have this new automated installer for OpenShift on bare metal. And that's definitely a trend that we're seeing is more customers not only looking at containers, but looking at running containers directly on bare metal environments. OpenShift provides an abstraction, which combines Kubernetes on top of Linux with RHEL, by really across all environments from bare metal to virtualization platforms to the various public clouds and out to the Edge. But we're seeing a lot of interest in bare metal. This is basically increasing the automation to install seamlessly and manage upgrades in those environments. We're also seeing a number of other enhancements, OpenShift service mesh, which is our Istio based solution for managing the interactions between microservices, being able to manage traffic against those services, being able to do tracing. We have a new release of that on OpenShift 4.6. And then some work specific to the public cloud that we've started extending into the government clouds. So we already supported AWS in Azure. With this release, we added support for the AWS government cloud, as well as Microsoft Azure government. And so again, this is really important to like our public sector customers who are looking to move to the public cloud, leveraging OpenShift as an abstraction, but wanted to support it on the specialized clouds that they need to use with Azure, the AWS. So Joe, can we stay there for a minute? So bare metal, we're talking performance there, because you really want to run fast, right? So that's the attractiveness there. And then the point about Istio in the OpenShift service mesh, that makes things simpler. Maybe talk a little bit about sort of business impact and what customers should expect to get out of these two things. So let me take them one at a time, right? So running on bare metal, certainly performance is a consideration. I think a lot of folks today are still running containers and Kubernetes on top of some form of virtualization, either a platform like vSphere or OpenStack or maybe VMs in one of the public clouds. But containers don't depend on a virtualization layer. Containers only depend on Linux and Linux runs great on bare metal. So as we see customers moving more towards performance and latency sensitive workloads, they want to get that bare metal performance and running OpenShift on bare metal and their containerized applications on that, a platform certainly gives them that advantage. Others just want to reduce the cost, right? They want to reduce their VM sprawl, the infrastructure and operational cost of managing a vert layer beneath their Kubernetes clusters. And that's another benefit. So we see a lot of uptake in OpenShift on bare metal. On the service mesh side, this is really about how we see applications evolving, right? Customers are moving more towards these distributed architectures, taking formerly monolithic or end-tier applications and splitting them out into lots of different services. The challenge there becomes then, how do you manage all those connections, right? Because something that was a single stack is now comprised of tens or hundreds of services. And so you want to be able to manage traffic to those services. So if a service goes down, you can redirect those requests to an alternative or a failover service. Also tracing, if you're looking at performance issues, you need to know where in your architecture you're having those degradations and so forth. Those are some of the challenges that people can sort of overcome or get help with by using service mesh, which is powered by STL. And then, I'm sorry, Stephanie, let me get to you in a minute, but just one follow-up on that, Joe. So the real differentiation between what you bring and what I can just... If I'm in a monocloud, for instance, you're going to bring this across clouds, you're going to bring it to on-prem and we're going to talk about the edge in a minute. But is that right from a differentiation standpoint? Yeah, that's one of the key differentiations. Red Hat's been talking about the hybrid cloud for a long time. We've been articulating our open hybrid cloud strategy. And even if that's not a strategy that you may be thinking about, ultimately where folks end up, right? Because all of our enterprise customers still have applications running in the data center, but they're also all starting to move applications out to the public cloud. As they expand their usage of public cloud, you start seeing them adopt multi-cloud strategies because they don't want to put all their eggs in one basket. And then for certain classes of applications, they need to move those applications closer to the data. And so you start to see edge becoming part of that hybrid cloud picture. And so what we do is basically provide a consistency across all those environments, right? We want to run great on Amazon, but also great on Azure, on Google, on bare metal in the data center, bare metal out at the edge, on top of your favorite virtualization platform. And yeah, that consistency to take a set of applications and run them the same way across all those environments is just one of the key benefits of going with Red Hat as your provider for open hybrid cloud solutions. All right, thank you. Stephanie, we'll come back to you here. So, I mean, we talk about RELL a lot because it's the business unit that you manage. But, and we're starting to see Red Hat's edge strategy unfold, kind of RELL is really the linchpin. I wonder if you could talk about how you're thinking about the edge and particularly interested in how you're handling scale and why you feel like you're in a good position to handle that massive scale and the requirements of the edge and versus, hey, we need a new OS for the edge. Yeah, I think, and Joe did a great job of setting up. It does come back to our view around this open hybrid cloud story has always been about consistency. It's about that language that you speak, no matter where you want to run your applications and between RELL on my side and Joe with OpenShift and of course, you know, we run the same Linux underneath. So RELL Core OS is part of OpenShift. That consistency leads to a lot of flexibility, whether it's through a broad ecosystem or it's across footprints. And so now as we have been talking with customers about how they want to move their applications closer to data, you know, further out and away from their data center. So some of it is about distributing your data center, getting that compute closer to the data or closer to your customers. It drives some different requirements, right? Around how you do updates, how you do over the air updates. And so we have been working in typical red hat fashion, right? We've been looking at what's being done in the upstream. So in the Fedora upstream community, there is a lot of work being that has been done in what's called the IoT special interest group. They have been really investigating what the requirements are for this use case and edge. So now we're really pleased in our most recent release of RELL 8, RELL 8.3. We have put in some key capabilities that we're seeing being driven by these edge use cases. So things like, how do you do quick image generation? And that's important because as you distribute what that consistency, create a tailored image, be able to deploy that in a consistent way, allow that to address scale, meet security requirements that you may have also, right? Updates become very important when you start to spread this out. So we put in things in order to allow remote device mirroring so that you can put code into production and then you can schedule it on those remote devices to happen with the minimal disruption. Things like, things like we all know now, right?  So we put in capabilities around being able to use something called RPM ostry in order to be able to deliver efficient over the air updates. And then of course you got to do intelligent rollbacks for per chance that something goes wrong, how do you come back to a previous state? So it's all about being able to deploy at scale in a distributed way, be ready for that use case and have some predictability and consistency. And again, that's what we build our platforms for. It's all about predictability and consistency and that gives you flexibility to add your innovation on top. I'm glad you mentioned intelligent rollbacks. I learned a long time ago. You always ask the question, what happens when something goes wrong? You learn a lot from the answer to that. But we talk a lot about cloud native. Sounds like you're adapting well to become edge native. Yeah, I mean, we're finding whether it's in the verticals and the very specific use cases or whether it's in sort of an enterprise edge use case, having consistency brings a ton of flexibility. It was funny, one of our, I was talking with a customer not too long ago and they said, agility is the new version of efficiency. So it's that having that sort of language be spoken everywhere from your core data center all the way out to the edge, that allows you a lot of flexibility going forward. Joe, I wonder if you could talk about, I mentioned just mentioned cloud native. I mean, I think people sometimes just underestimate the effort it takes to make all this stuff run in all the different clouds, the engineering efforts required. And I'm wondering what kind of engineering you do if any with the cloud providers and of course the balance of the ecosystem, but maybe you could describe that a little bit. Yeah, so Red Hat works closely with all the major cloud providers, whether that's Amazon, Azure, Google or IBM cloud, obviously. And we're very keen on sort of making sure that we're providing the best environment to run enterprise applications across all those environments, whether you're running it directly just with Linux on RHEL or whether you're running it in a containerized environment with OpenShare, which includes RHEL. So our partnership includes work we do upstream. For example, Red Hat helped Google launch the Kubernetes community and I've been with Google, we've been the top two contributors driving that project since inception, but then also extends into sort of our hosted services. So we run a jointly developed and jointly managed service called the Azure Red Hat OpenShift Service together with Microsoft, where our joint customers can get access to OpenShift in an Azure environment as a native Azure service, meaning it's fully integrated just like any other Azure service you can tie it into Azure billing and so forth. It's sold by Azure or Microsoft sales reps. But we get the benefit of working together with our Microsoft counterparts in developing that service and managing that service and then in supporting our joint customers. We over the summer announced sort of a similar partnership with Amazon and we'll be launching or already doing pilots on the Amazon Red Hat OpenShift Service, which is the same concept now applied to the AWS cloud. So that'll be coming out GA leader this year, right? But again, whether it's working upstream or whether it's partnering on managed services, I know Stephanie and team also do a lot of work with Microsoft, for example, on SQL Server on Linux dot net on Linux, whoever thought you'd be running that net applications on Linux, but that's a couple of years old now or a few years old. So again, it's been a great partnership not just with Microsoft, but with all the cloud providers. So I think you just shared a little leg there, Joe. What's coming GA later this year? I just wanna circle back to that. Yeah, so we announced a preview early this year of the Amazon Red Hat OpenShift Service. It's not generally available yet. We're taking customers who wanna sort of be early access to pilots and then that'll be generally available later this year. Although Red Hat does manage our own service, OpenShift dedicated, that's available on AWS today, but that's a service that's solely operated by Red Hat. This new service will be jointly operated by Red Hat and Amazon together. It'll be sort of a service that we are delivering together as partners. As a managed service, and okay, so that's in beta now, I presume, if it's gonna be GA later this year. Yeah, it's like a, yeah, yeah. And that's probably running on bare metal, I would imagine. That one is running on EC2, so that's running on AWS EC2, exactly. Can I run it on bare metal? If I wanna turn it over. All of our, I mean, that OpenShift does offer bare metal cloud and we do have customers who can take the OpenShift software and deploy it there. Right now, our managed offering is running on top of EC2 and on top of Azure VMs. But again, this is appealing to customers who like what we bring in terms of an enterprise Kubernetes platform, but don't wanna operate it themselves, right? So it's a fully managed service. You just come in, build and deploy your apps and then we manage all of the infrastructure and all the underlying platform for you. That's gonna explode, my prediction. Let's take an example, a hard example is security. And I'm interested in how you guys ensure a consistent security experience across all these locations, on-prem, cloud, multiple clouds, the edge. Maybe you could talk about that. And Stephanie, I'm sure you have a perspective on this as well from the standpoint of REL. Who wants to start? Yeah, maybe I could start from the bottom and then I'll pass it over to Joe to talk a bit. I think one of these aspects about security, it's clearly top of mind of all customers. It does start with the very bottom and base selection in your OS. We continue to drive SE Linux capabilities into REL to provide that foundational layer. And then as we run REL core OS into OpenShift, we bring over that SE Linux capability as well. But there's a whole lot of ways to tackle this. We've done a lot around our policies around CVE updates, et cetera, around REL to make sure that we are continuing to provide and commit to mitigating all criticals and importance, providing better transparency to how we assess those CVEs. So security is certainly top of mind for us. And then as we move forward, there's also, and Joe can talk about the security work we do, there's also capabilities to do that in containerization. But we work all the way from the base to doing things like these images and these easy to build images, which are tailored so you can make them smaller, less surface area for security. Security is one of those things that's a lifestyle, right? You got to look at it from all the way the base and the operating system with things like SE Linux to how you build your images, which now we've added new capabilities there. And then of course in containers, there's a whole focus in the OpenShift area around container security. Joe, anything you want to add to that? Yeah, sure. I mean, I think obviously Linux is the foundation for all public clouds. It's driving enterprise applications in the data center. Part of keeping those applications secure is keeping them up to date. And through RHEL, we provide secure and up to date foundation as Stephanie mentioned. As you move into OpenShift, you're also then able to take advantage of essentially immutability, right? So now the application that you're deploying is an immutable unit that you build once as a container image. And then you deploy that out to all your various environments. When you have to do an update, you don't go and update all those environments. You build a new image that includes those updates and then you deploy those images out in a rolling fashion. And as you mentioned earlier, you can go back if there's issues. So the notion of immutable application deployments has a lot to do with security and it's enabled by containers. And then obviously you have Kubernetes and all of the rest of our capabilities as part of OpenShift, managing that for you. We've extended that concept to the entire platform. So Stephanie mentioned RHEL CoreOS. OpenShift has always run on RHEL. What we have done in OpenShift 4 is we've taken an immutable version of RHEL. So it's the same Red Hat Enterprise Linux that we've had for years. But now in this latest version RHEL 8, we have a new way to package and deploy it as a RHEL 8 CoreOS image. And then that becomes part of the platform. So when customers want to, in addition to keeping their applications up to date, they need to keep their platform up to date. So they need to keep, up with the latest Kubernetes patches, up with the latest Linux packages. What we're doing is delivering that as one platform. So when you get updates for OpenShift, they could include updates for Kubernetes. They could include updates for Linux itself as well as all the integrated services. And again, all of this is just, this is how you keep your application secure is making sure you're taking care of that hygiene of managing your vulnerabilities, keeping everything patched and up to date and ultimately ensuring security for your application end users. I know I'm going a little bit over, but I have one question that I want to ask you guys. And it's a broad question about, maybe it's trends you see in the business. I mean, you look at, we talk a lot about cloud native and you look at Kubernetes and the interest in Kubernetes off the charts. It's an area that has a lot of spending momentum. People are putting resources behind it, but really to build these sort of modern applications, it's considered state of the art. And you see a lot of people trying to really bring that modern approach to any cloud. We've been talking about Edge. You want to bring it also on-prem. And people generally associate this notion of cloud native with this kind of elite developers, right? But you're bringing it to the masses. I mean, there's 20 million plus software developers out there. And most, with all due respect, they may not be the elites of the elite. So how are you seeing this evolve in terms of re-skilling people to be able to handle and take advantage of all this cool new stuff that's coming up? Yeah, I can start. OpenShift, our focus from the beginning has been bringing Kubernetes to the enterprise. So we think of OpenShift as the dominant enterprise Kubernetes platform. Enterprises come in all shapes and sizes and skill sets, as you mentioned. They have unique requirements in terms of how they need to run stuff in their data center and then also bring that to production, whether it's in the data center across the public clouds. So part of it is making sure that technology meets the requirements and then part of it is working the people process and culture to help them understand what it means to sort of take advantage of containerization and cloud-native platforms and Kubernetes. Of course, this is nothing new to Red Hat. This is what we did 20 years ago when we first brought Linux to the enterprise with RHEL. And in essence, Kureza is basically distributed Linux. Kubernetes builds on Linux and brings it out to your cluster, to your distributed systems and across the hybrid cloud. So nothing new for Red Hat, but a lot of the same challenges apply to this new cloud-native world. Awesome. Stephanie, we'll give you the last word. All right. And I think just to touch on what Joe talked about, it's, and Joe and I work really closely on this, right? The ability to run containers, right? As someone launches down this, because it is magical what can be done with deploying applications using a container technology, we built the capabilities and the tools directly into RHEL in order to be able to build and deploy, leveraging things like Podman directly into RHEL. And that's exactly so, folks, everyone who has a RHEL subscription today can start on their container journey, start to build and deploy that. And then we work to help those skills then be transferable as you move into OpenShift in Kubernetes and orchestration. So, you know, we work very closely to make sure that the skills building can be done directly on RHEL and then transfer into OpenShift because as Joe said, at the end of the day, it's just a different way to deploy Linux. Guys, you're doing some good work. Keep it up. And thanks so much for coming back in theCUBE. It was great to talk to you today. Good to see you, Dave. Yeah, thank you. And thank you for watching, everybody. theCUBE's coverage of KubeCon NA continues right after this.


Hey guys, my name is Stefan today. We're talking about some competitive gaming some e-sports and some call of duty Let's start with call of duty first as that is the biggest topic of this video So call of duty ghosts last big proudly competition is coming up at UMG Nashville October 10th And we want to send a team out there to represent So how would we do that? Very very simple. Go check out the link in the description below Fill out the form we'll get back to you Very very easy. I also want to talk to you about a new YouTube channel We've opened up for competitive gamers YouTube comm slash e-sport go check it out. Go subscribe There's not that many videos up there on there quite yet But we want to make sure that that is going to be the home of e-sport on YouTube And we will be pushing for that in the near future to help with that We are actually going to be at UMG Nashville ourselves freedom will be there We will be represented I'll be there a few other freedom staff members will be there And we want to make sure that you're there too. So tell us if you're gonna be at UMG Tell us if you'd like to be at UMG And of course tell us other events to do either with gaming or YouTube that you think that freedom should be represented at We would love to make sure that we can meet as many of you throughout the next year as possible But we need to know where will you guys be? Do you guys attend events? Do you think events are a waste of time? We want to know all these things in the comments below. So I'll keep you guys updated with the teams coming through here for UMG and I will also make sure that you guys Hear more about me when it comes to competitive gaming and I'll talk to you guys soon. Goodbye


Okay. Good afternoon. So I'm, as you already know, I'm Götze and I'm giving you the course in algebra. So my room is 110 and one of our postdocs who's called Tarek Abdel-Gadir. I'm not sure. So he said that if his room is 122, so if you have questions that you do not want to ask to me or whatever or anyway, if you just have, you can also talk to him. He would like also to see you anyway. I'm usually in my room whenever you have questions about what I say you can come either to me or to him. And okay. So the topic of this course is algebra. And we will be somehow have three parts. We'll be dealing with groups, rings, and fields. In the case of the fields, we'll do Galois theory also. So this course essentially starts from zero. So, I mean, we start with the definition of a group. So it's quite elementary. Maybe for some of you it will be boring. I mean, if it's really boring for everybody, we can kind of change it. But on the other hand, if it's only boring for some of you and some others actually are learning something, we keep it the way it is. And so I will just start by the definition of group. If you have any problems, if I'm going to fast or you don't understand something, you tell me. If you cannot read my handwriting, you also tell me, but I can not permanently improve it. You know, you will also have to learn to read it. Okay. So let's start. So what's the group? By the way, you have my notes. I have prepared some notes. I will actually be following the notes very closely. I've just prepared them before starting this course. They are so new that they are also still full of misprints, as you will have maybe noticed. But you know, anyway, I will, I hopefully will say the correct things in the lecture. So a group will be a set, will be with a binary operation. So somehow you can multiply two elements, A, B. You can multiply them and get an element A times B. So somehow you know, you, so it somehow is set in which you can multiply the elements and some obvious actions should hold. There should be the associative law. So it doesn't play a role how you put brackets for this operation. And you have a neutral element and every element has an inverse. So let me write this down properly. So definition. So we take a non-empty set, set G with the binary operation. So this is multiplication from G times G to G, which sends a pair of elements AB to something which we want to call A times B. So just, you know, binary operation just means that for two elements we get one element is called the group. If the following actions are true, well, first we have the associative law. So we can, so in, it's written like this. So if you have A times B, so for, this is always for all A, B, C elements in G. If we take A times B and we multiply this by C, this is the same as if we multiply B with C and on the other side, multiply it by A. So this just means, you know, we have this operation. So A times, so we can, it just means we first do this operation. We multiply them, get an element of G, we multiply with this, and then we do this with the binary operation. Yeah, this you, it is in principle, so binary is, with the binary operation. So I will try to maybe write bigger. So this is the first thing, and I mean, you know, it's kind of a standard thing for any kind of operation that you need is a stativity. It's the first thing that you always require. Second one, we have a neutral element. So that means there exists an element E in G such that we have that E times A is equal to A times E is equal to A for all A in G. So it's an element for which the multiplication, it doesn't do anything. And the last one is that every element has an inverse so that if you multiply the element by the inverse, you get the neutral element. So for every element for all A in G, we have an element A to the minus one in G such that A times A to the minus one is equal to A to the minus one times A is equal to E. Okay, so these are the simple actions. And it somehow captures some part of what we used to if we kind of multiply elements in Z or in the complex numbers, the real numbers or the rational numbers, or if we add integers. And so we will, we can by this definition, we can simplify slightly the notations. For instance, as the bracketing doesn't play any role, we can also forget the brackets. So we write also ADC for, so first we write AB for A times B. So we usually will kind of suppress the operation and we can also write ADC for ADC, which is the same as ADC. After all, it doesn't play a role. It's verified. So we will start. Now I want to, as an exercise in these definitions, which are kind of straightforward definitions, so just abstractly use the definitions. We want to prove some elementary properties of groups, but really elementary. So this should seem completely trivial to you. So we have remark. Let's say first the neutral element is unique. So as I said, this is not really, these are not particularly great results, but the point is that we have here our actions and we want to argue only the actions and you have to get familiar with these abstract arguments if you are not already. So let E and E prime be neutral elements. So that means if I multiply anything by E, I get back the other thing I multiply by. So we have this property. So then while we can say E is equal to E times E prime because after all, E prime is also neutral element and this is equal to E prime because E is a neutral element. And so it's equal to E prime. Okay. So the neutral element is unique. Any two neutral elements are equal. And then, so this was the first one. Second one is, so the inverse is unique. That's somehow similar. So let N prime the inverses of A in element in G. Well, so we can write B prime, same. We want it there as B times A, E. E is our neutral element. So this is B prime times A, D because E is inverse to A and this is equal to D prime. So this was like this. Now we can use the associativity. So this B prime A times B. This is E equal to B. Okay. So it's really straightforward. Then we've just used the actions. The third statement would be, so it's enough to be an inverse. Actually it's also true for a neutral element. It's enough to be that one of these two identities holds. So that it is a left inverse thing or left identity. But now we have to say it for identities. So let B in G be an element with B times A is equal to the identity element. Then it is the inverse. Where this one can do in different ways. But I can also just, I mean, I would just write like this if I take the inverse. So I'm actually, then I can write this as E times A to the minus one. And E I can write as B A and as A is the inverse. This is equal to B. Okay. So these are all very simple things. And then maybe I don't do the other ones which are similar. I mean, they are even simpler. So maybe fourth one is that if I take the inverse of A to the minus one, this is A. Well, this is actually almost total logical. We know that A times A to the minus one is equal to A to the minus one times A equal to E, which says precisely that A is the inverse of A to the minus one. And then finally, as one uses it often, if I take the inverse of the product, this is the product of the inverses, but the other way around. Well, and that's in some sense clear. If I take A B times E to the minus one, A to the minus one, then this is, you know, A B E to the minus one, A to the minus one. So I first can cancel these. This is A, A to the minus one, and this is E. So they are inverse to each other. Obviously, okay. So much for that. So this was very simple. So there's another property which is also useful, which is sometimes called the cancellation property. So that is, if I have an entity in group, so let's again A, B, C, the elements in G, and assume that A, B is equal to A, C, then it follows. So if, then it follows that B is equal to C. So you can always cancel equal factors. And in the same way, if B A is equal to C A, and also for that B is equal to C. And this is obvious. You just multiply with the inverse of A on both sides. Here you multiply with A to the minus one on this side, get B equal to C. So if I take A to the minus one, A, B, then before they were equal, now I multiply. This will be equal to A to the minus one, A, C. And I can forget the brackets as usual. And then this is E. So this is equal to B. And this is equal to C. And the same way for the other equation. So there's another property that we are very used to with the groups that we know is that many groups are commutative. So it doesn't play a role in which order the elements are when we multiply them. This is true for the integers. This is true for the rational numbers and so on. So I will, as although you all know it, I will still kind of write it down. So definition, a group G, G is called commutative if, well, if I take A times B is equal to B times A for all A and B in G. And I want to also introduce notation for that. So the kind of standard, it is most of the time when you have a commutative group, not always, but most of the time you write it in additive notation. So you write plus instead of times. And so often for commutative groups, so write the operation as plus. So this then says we write, we write A plus B instead of A times B. And we write minus A instead of A to the minus one. And we can write A minus B instead of A times B to the minus one and we write zero for the neutral element. So this is most of the time when we have a commutative group we will do that, but not always. So now I want to give you a few examples which you essentially all know to see that groups are all over the place. And so the first example, I mean, anyway, first are there any questions until now? I mean, until now I've just written here these definitions and have some simple computations. So if there's anything which is not clear at this point, you should really tell me because then I have to change my approach. But if it should feel kind of trivial to you, but you know, you shouldn't be extremely super bored, but you know, okay. Any comments? No. Okay. So can you now essentially read what I write or is it very difficult? It's okay. What? No, no. What? Yes.  We have a binary operation which satisfies certain properties and then we call, we denote it either by times or by plus. But you know, it's not, you know, you can obviously even on the integers you could define a binary operation which has nothing to do with the addition or the multiplication. You know, it's just that we call it like that. We call, in this case, we call it sum, but it doesn't have to mean the sum. You know, it just is a notation that we put. I will give some examples although in that case it actually is the sum. Anyway, so first we give some examples. So we first can take, for instance, the first example is the trivial group. So a group should be a non-empty set. So in particular, the smallest non-empty set consists of one element. We could, for instance, take the element which consists of an element which we call one or E and this becomes a group by defining that E times E is defined to be equal to E and then you can check that this is a group. Obviously, E is the neutral element and it's also its own inverse. Okay?  So we can take, for instance, the integers set with the usual addition z plus. So really an integer is an integer and you just add them the usual way. This will be in a b. Everybody knows that this is the case, that you can add integers and the order which you add them doesn't play a role. Then, on the other hand, for instance, if we take z together with the multiplication, this is not a group. So again, we take the standard multiplication on the integers. This is not a group because not every element has an inverse. For instance, two has no, so you find that the neutral, so it has a neutral element. Obviously, it's one, but the inverse, so two, for instance, has no inverse. Actually, that was even a bit worse. I take maybe z without zero. It's not a group, otherwise it's even more wrong. I mean, also z obviously is not a group, but in that case you don't even have a neutral element. Then maybe we forget about this. So let me have a slightly more complicated example. So let k be some integer, say positive integer. So by this I mean that I want to define a group structure on the set of all integers from zero to k. So let me write although it's not standard. So I just write zk for the set of the element zero one until k minus one is set with k elements. And we want to make this into a group. And in different guises, we will kind of encounter this many times. So I do it like this. So if we have an integer, so if n in z is an integer, I denote by, I mean that's maybe not standard anyway, but I say n underlined the rest of division by k. So the rest of dividing n by k. So that is, we write n equal to d times k plus n bar, where d is an integer and n is an element of this set. And you learn in high school that you can always do this and this is the rest of this division. So now we want to make this. So then we define a group structure. So we define an addition on the set. So an operation which we call addition on zk, which I denote is like a different way. So I say maybe plus. So I say n. So if you have two elements here, n, which are elements of zk. So these are numbers from zero to k minus one. I can add them, I say, and I denote the addition like this. This is done by taking the usual sum as integers and then taking the rest under division by k. So if I have two numbers, which lie between zero and k minus one, if I take the sum, it doesn't necessarily lie between zero and k minus one, but I can, if it's too large, I can subtract k from it and I get an integer in here. And this I define to be the sum. And I claim this is an nabian group. This makes this thing into an nabian group. So that k with this operation plus is an nabian group. Well, this is kind of trivial. So we have to check the action. So we, for instance, about the sensitivity. So what we have to prove is that if we take such a triple sum m plus n plus l, where these elements zk, this is the same as doing it the other round. We have just to look at the definition. So this is we take n plus m, the usual sum, and the rest under division by k. And then we sum this. So maybe for the moment I write plus l, which is by definition n plus m plus k plus l. And so, you know, but this just means we, you know, we sum these elements and we subtract the corresponding multiple of k until we get in the elements at k. So this is just the same as the sum of all three rest under division by k. And this now is symmetric. So if we can kind of perform this operation backwards, this is equal to n plus m plus l equal to, well, that's what I wanted. I mean, you just make these steps on the other side backwards and you get this. Because this is already completely symmetric and you know that in the, so this would be first, if you write like this, it would be first like this. But you know the sensitivity holds in z. So you can also, so maybe you write it like this. This is equal to n plus m plus l.  And the commutativity is also clear. You know, you know that if you take the sum n plus m, this is n plus m, the rest of this, but the addition in the integers is commutative. So this is m plus m. And obviously the neutral element for this addition is the element 0. Because if we add 0 to element in z, you get back the same element. It doesn't change when we take the rest under division by k. And the inverse of some element l, k minus 1, is okay. If l is equal to 0, we take 0. And if l, and otherwise we take the inverse of another l, which is different from 0, otherwise the inverse of l is k minus l.  Okay, so this is a kind of, it's a very simple example, but somehow we have, you will see that this can be viewed as an example of taking a quotient group of a group by a normal subgroup. The moment we don't need that to know that. So another, I give one more example, if I, so a bit more abstract, which you know from linear algebra. So let, I don't know which number I'm now at, but anyway, five. Let v be a vector space. Then I can call, say the, maybe the group of automorphisms of v is a set of all, say, v from v to v, where v is a linear map. So vector space, say, r vector space or something does matter. v is a linear map. So phi is linear and phi is projective. So then v is from a group, so out of v is a group. Yeah, well, I can take any vector, any field, but you know, whatever, I could say the field, say v is an r vector space, and then this would be r linear, but you know, whatever, no, any field. So out of v is a group. Well, you know, you know that, so the group operation is a composition of maps. So I can say that, so phi times c would be phi composed with c. I know they are both projective maps. I can compose them and I get another projective linear map. And so, so we know it's a standard fact that one learns early in the one's life that composition of map is always associative. So this is associative from the definition just by remembering what it means to compose map. And we see that if, so if I take the identity of v from v to v, which sends any vector to itself, is a neutral element. Because if I compose with the identity, I get back the same map. And you know, if phi is an automorphism of v, it is a projective map. So it has a projective linear map. So it has a projective linear inverse. So we have phi to the minus one v, the inverse map. And then it is also the inverse in the group. So phi to the minus one is the inverse element in, out of v. Because by definition, to be the inverse map precisely means that both compositions of phi and phi to the minus one are the identity. phi composed with phi to the minus one is equal to phi to the minus one composed with phi is equal to the identity of v. That's what it means to be the inverse map. And that's the same definition as what it means to be the inverse element in the group. Okay. A similar thing that we can look at is just, you know, the symmetric group on a set, which is a more general statement. So if, so six, let m be any set. We can just call, so the group s of m is just defined to be the set of permutations of m is just a set of all bijections of m to the set. And by the same token as before, I mean the same argument will actually see that this is a group by composition. So that means, so again, phi times c is equal to phi composed with c and the composition of bijective maps is a bijective maps. And we have again the identity of m is a neutral element and the inverse map to a given bijection is the inverse element. Okay. So this is, I mean somehow okay. So we have many groups. This is not, so traditionally when groups were introduced, they were always viewed as groups of permutations of some set. So they were always thought of being that. That's how groups were first considered. I mean, this was what the group was supposed to be. And so in that sense, this kind of goes back to the standard things. And actually, this comes up also in the context of Galois theory, which we will do at the very end of our course. In fact, groups were invented to do Galois theory, which is, you know, somehow. Anyway, so we have here. So we look at a special case of this when we have a kind of very simple set. So if m, we can look at the symmetric group on just the set of elements one to n. So on this finite man is called the symmetric group n. And I will usually denote it, I think, by, I don't know what my notation is. Maybe I will just call it sm. Okay. Usually we have some kind of Gothic s or something, but as I cannot write that, I write like this sm. This is, so it's a special case of this where the set m of which we take the permutations. So all the bijective maps to itself is a finite set, one to n. And here one can be very explicit. So again, the group operation is just a composition like this. And we want to make some computations in this. So what, yes? Yeah, okay. No, it has nothing to do with the order of the group. Actually, the order of the group is much bigger.  So maybe one could also say the symmetric group, sometimes I would call it symmetric group on n letters. So somehow you have n letters and you permute them. Maybe that's a better name. Okay. So we introduced the following notation for such a permutation. So if I have an element, so for you write an element f from, so I've got it in sn, so it's a bijection from the set one n to itself. In sn, we write this as, so we write first all the elements one to n, one, two, three, and so on until n. And below we write their image. So f of one, f of two, f of n. Okay. So we have some notation how to write this thing. Later we will also find other notations. So let's look at, so we can look at some examples. For instance, s two, all the bijections from the set one, two to itself, they're not very many. There's, it consists of two elements. There's either the identity element, so the identity element, which is just, so it consists of one. So the identity, which I, in this notation would be one, two, one, two. And there's only one more element, which is permuting the two. So one, two, two, one. So this has two elements. We can also look at s three. Again, we start with the identity element. So, so every element goes to itself. And then we have to look at all possibilities what we can do. We can first permute these two. So this is, there's left second to it's one, two, three, one, three, two. Then we can make this go to two, one, two, and skip the order of the other ones. So two, three, then we can exchange these two. Then we have exhausted everything we can do if one goes to two. And then we can get one goes to three. We can keep the order of the other one. So we have two, three, which goes to one, two, or we can exchange the order of the other ones. And you can easily convince yourself that these are all elements. Maybe it could be, and so how does one, and then this is our, I mean, actually maybe in the, in this case, we can see that the group is not commutative. So, so we've seen that this is a group because the special case of this group, but it's not commutative. So s three is not commutative. And we can see this by seeing what happens if we multiply two elements. So for instance, we can look at these two elements. One, two, three goes to one, three, two. And we multiply it by one, two, three goes to two, one, three. Yeah. And so what does this give to us? So we have to remember that the operation was composition. So in this case, we have here one goes to two, and then two goes to three. So one goes to three. Here two goes to one, and then one goes to itself. And here three goes to three, and then three goes to two. So this is how, what this gives. And we can also do it the other way around. Do the same thing. So one goes to one, and then one goes to two. You can already see it's not commutative. So two goes to three, and then three goes to itself. And finally three goes to two, and then two goes to one. So we see it's not commutative. What? Yeah. Yeah, that's true.  Now that's correct, yeah. But I mean, I, it's not, okay. You have to, some books on algebra do this, and some don't. I mean, it's not like everybody doesn't do it. I'm kind of going against the, so obviously in all other fields of mathematics, which is not algebra, one does it in this way. So in algebra, one sometimes also does the other way around. But you know, I kind of, I prefer to do it the way, which is kind of general.  But it's kind of equivalent. No, it's not a big difference. And normally the composition of maps is always done in this way, although it is somehow slightly, you know, you know, it feels maybe slightly wrong. Yeah, but this again is only done in group theory and not in any other field.  I mean, it's not, even if in group theory, sometimes one does a different thing. But you know, you're free to do it the other round. What? It's confusing for you or? Yeah. Well, but the different, there's no real big difference. So, and otherwise it's confusing for me.  I mean, okay, no, no, but you're right. It is very often, so in group theory you find that, but it's kind of strange. I find it, you know, that there's one small group of mathematicians who does it different from everybody else is somehow also wrong. So, yeah, so if you want, so an exercise could be, I mean, and this is not homework, but I'm just saying if it should be easy to check that if you look at the number of elements, so actually, so I, so the number of elements of Sn is equal to n factorial, so for, so n factorial as you, I hope, for knows is just the product of all numbers up to n. So, maybe I should say in this context, as this also already came up, so if G is a group, the order of G is, well, it's just the number of elements of G, so it's just this number, but then one sometimes also denotes it as in a different way is order of G, which is just G, the number of elements of G. So, we write the order of G is infinite, so equal to infinity if G is infinite, so if it's not a finite group, once the integers have this, and so here the claim would be that the order of Sn is n factorial, well, maybe I don't do this example. Okay, so one more piece of notation, so it's kind of useful to have some kind of notation like one uses it for the integers that one can look at powers of elements in the group and things like that, and so we just introduce this, so if G is a group and A an element in the group, in the group, then we, and we have an integer, so first we want to define A to the n for an integer n in Z, this is just supposed to be an element in G, so if n is bigger than zero, so a positive integer, we let A to the n just be the product of n times element A, A times, times A n times, okay? If n is equal to zero, A to the zero, we define to be the unit element E, E is the neutral element, so this is our definition, and if n is, say, bigger than zero and we take A to the mean of minus n, so we take a negative power, so if n is smaller than zero, we say that A to the n is defined to be the inverse of A to the power n, well, maybe minus n, okay? So this defines, gives a definition for A to the n for all elements in Z, so this is no problem, and we obviously, in order for this to be making any kind of sense, we have to know that kind of the standard rules that you know for, you know, the usual numbers when taking power still apply, so this I formulate as an exercise, so show that for any element A, we have A to the, so how shall I put it? A to the n times A to the m is equal to the A to the n times m for all n and m integers, and if we take A to the minus n, this is the same as A to the minus 1 to the n, or maybe I want to do the other round, this was a definition, so this is A to the n to the minus 1, okay? And this is basically, or again, a simple exercise in the definitions, you know how to add and multiply integers, so these are just integers and then you have to see that you, that this always holds, and this again for all n and z, so we can write this power kind of notation, okay? So much for the, you know, the very basic things about groups, now we want to, are there any questions under here? Now we want to introduce subgroups, so these are, you know, subgroups, subsets of a group which, you know, if I restrict the group operation of the bigger group to the smaller set, the smaller set becomes also a group, so let's do that. So just to make our life a bit easier, I will, from now on, we usually denote by one the neutral element of our group, so not always, sometimes we might denote it by zero, or sometimes we might still denote it by E, but whenever I write one, it will be the neutral element of our group, and in fact to just say, so if this were already had, if our, if G is a billion, and we use the additive notation, then I already had said that we still denote the neutral element as zero, no? So here if you write a plus b for the group operation, then we denote the neutral element as zero. So let's do the subgroup, so definition, so as I said it's a subset of a group, such that with the operation of the group it becomes a group itself, so let G be a group, H subset G is subset, so H is a subgroup of G, well first I just say what I just said, so if first for any two elements a and b in H, we have that a times b is also an element in H, so here we have the operation in G, and the operation in G has the property that it will send, and we want that it has the property that if you have two elements in this subgroup, the product also lies there, and the second one is that with this operation H is a group, and with this operation, so with the restriction modification from G times G to H times H, H is a group. Well strictly speaking it follows from this, but because we know that the group is non-empty, but obviously I can say that it should be a non-empty subset. What? Which condition? No, no I'm actually asking this, I'm making, in the moment this is maybe not the most useful definition, but it is the most natural definition, a subgroup of a group is a subset which with the operation of the group is a group, of the MN group is a group, this is the definition, and then obviously that's what I'm saying here, so I'm saying the product of any two elements must be there, and then if I take this as operation it is a group, that's the requirement, so that's a definition, but you are right that this is usually not how one states it, because what one usually states is a criterion to be a group, and that I can also write, so we have the following criterion, so lemma, so let G be a group, and H, so a non-empty subset H of G is a group, is a subgroup, well if we have this property and also another one, so if for all, so if we have that A times B is in H for all A, B in H, and A to the minus one is an element in H for all A in H, there's another way of saying, I mean anyway that you would see, okay, so this is, if you want often you find this also as a definition, let's see why this is equivalent, so we have to see that, so the first condition is anyway the same, we have to see that with these two conditions, H will actually be a group with this operation, so first and conversed, so obviously if H in G is a subgroup, then these two conditions hold, because for a group we have an operation on the elements in the group, so the product of any two elements lies in the group, and the inverse of elements also must lie in the group, because it's a group, so then, so maybe I write this one and two, so one and two obviously hold, okay, so conversely, but it's not so difficult, so the first one, so we now want to show it's a group, so what we have to show, I mean first we have to show the laws of that it's a group, so if for A, B, C in the group G, we have the associative law, so thus it also holds in H, because if I have three elements in G, then if I have three elements in H, then in particular elements in G, and so the associative law holds for them, thus it also holds for A, B, C in H, so this is the first extreme, and the second one is we need to have a neutral element, so if, so let A be an element in H, then it follows that A to the minus one is an element in H by this extreme, and we have that A times A to the minus one is equal to the identity element is equal to A to the minus one times A, and this shows two things, so as a, so so it follows that as the as a product of two elements in H, one is an element of H, because this first one here says that the product of any two elements in H lies in H, so and H, one is a neutral element of G, so it's also a neutral element of H, because being a neutral element means if I multiply anything by it, it stays unchanged, so follows that H has a neutral element, and on the other hand this thing we also see here that A to the minus one is the inverse element of A, so and A has an inverse, so we see this is a kind of trivial reformulation, I mean if one wants one can also, I don't, I never found this particularly useful, but you often find this, it's an easy remark that a non-empty subset H subset G is a subgroup, if we can just write this in some sense in one formula, if A times B to the minus one is in H for all A in H, so if you want to go and call this exercise, it's easy to see that this is enough to imply this, you can check that, so now let's look at some examples of subgroups, so for instance, so we have z, we look at z where there is an addition, so integers with standard addition, and so if k is an integer, we can look at, let's say kz to be the set of all integers which are multiples of k, so this is a set of all n times k where n is an integer, maybe I should write it in the same order, doesn't matter as it's commutative, so these are all integers which are divisible by k, so then I claim this is a subgroup and I expect you can, I mean it's kind of clear that, so here we have the additive notation, so the neutral element of z is the element zero, and the element zero is contained in this, and well anyway, so we just have to see that if you take a product of two integers divisible by k, then if you take a sum of two integers which are both divisible by k, then the sum is also divisible by k, and if you take minus an integer which is divisible by k, it's also divisible by k, so if you see anything to write, so if a comma b in are divisible by k, then so is a plus b, and also minus a, and so we precisely, now we have here this additive notation instead of times we have plus, instead of a to the minus one we have minus a, but we see that precisely these two things are fulfilled, okay these are kind of stupid, so maybe if h is a subgroup of a group g, and l is a subgroup, and l is a subgroup of h, then we find that l is also a subgroup of g, this is trivial from the definition, I mean or from this criterion, and it's also so if if say h and l are subgroups of g, then h intersect at l is a subgroup of g, again this follows directly definition if you all buy this criterion, so any the product of any two elements in h lies in h, and the product of any two elements in l lies in h, so if the elements before were both in element in h, then also a product is both in element in h, and similarly for the inverse, anyway you can check it, if it's not obvious to you it's an exercise, then one very special case of a subgroup, so I don't know which number we're at, maybe three, so let g be a group, and a an element in g, then we can look at the cyclic subgroup of g generated by a, so let a the subgroup generated by a, this is just a set of all powers a to the n, where n lies in z, so we take any number of products of a or any number of products of a and a to the minus one, so note we have this if one remembers this exercise that I formulated about these powers, we know that a to the n times a to the m is equal to a to the n plus m, so which means that the product of any two elements in this a again lies in a, and we know that if we take a to the n to the minus one, this is the same as a to the minus n, so this says that the inverse of any element here lies in a, so contains all products, and here contains inverses, so this means it's a subgroup, so we have a for any element we have a subgroup generated by it, and maybe one can check it's easy to see that a is a billion, so commutative, and I didn't write before what one also says a billion for this for that, a billion actually will be large, so it's just because you know after all the addition in z is a billion, so a to the m times a to the n will be a to the m plus n, which is the same as a to the n plus m, okay, so we get this a billion subgroup of our group, so we call a is called the cyclic subgroup of g generated by a, so example we had this already, if we take the group k times z, this is after all the set of all k times n with n in z we had it before, so this as you can see now we have here the additive notation, so the instead of looking at these powers we take the products, so maybe I should, so this is the cyclic subgroup of z generated by k, so maybe I should kind of also I'm not sure I introduce the notation, so if g, if we have the additive notation for the group, so if we write a plus b for the group operation, then as I said we write zero the neutral element and we write minus a the inverse of a and we would write for n in z we write n times a for what otherwise would have been called instead of a to the n, no, this is instead of a times b, this is instead of one and this is instead of a to the minus one, which does notation, so then you can see that this is the cyclic subgroup generated by k and what time yeah okay, so we will call a group cyclic if it's it looks like that, so definition the group g is called cyclic we write here if it is equal to the cyclic subgroup generated by one of its elements, so if there exists an element a in g such that a the subgroup generated by a is equal to g, so for instance we see that that z is cyclic, so for instance we have it generated, so z with the addition generated by the element one, so any integer can be obtained as n times one for some n that's kind of and we have if we take this group z g, z k which was just the set of elements zero to k minus one for k positive integer and this is also cyclic also generated by one because by definition one plus one is also here equal to two and so on until k minus one and k minus one plus one is equal to zero, so let me see maybe finally I talk about the subgroup generated by several elements, so definition let g be a group and say let u the subset the subset of g, so the subgroup generated by u is what one would call the smallest subgroup of g which contains u, so technically this means is the intersection of all subgroups say h of g which contain u, so obviously g does contain u, so there are such subgroups and we can take the intersection of all of them and we know that the section of two subgroups is again the subgroup and it's easy to see that this also holds for the intersection of any number of subgroups and so we have this we have this subgroup this is called the subgroup generated by u, so this is so in particular for finally many elements if say g1, g k are some elements of g we can consider the subgroup generated by them, so this is just I mean it's denoted maybe like this and it's just defined to be the subgroup generated by the set of these elements, so this is the smallest subgroup of g which contains g1 to gk, you can check as an exercise but it's a kind of standard that if we look if g is equal to z with the addition then if I take the subgroup generated by 4 and 6 this is equal to the subgroup generated by 2 which after all is equal to 2 times hope that's correct yes and you can easily see how you might want to generalize this to any numbers okay so that's maybe enough for for today, so I mean I don't know you can maybe you didn't make very many comments in between so I hope it was clear to you and you can maybe can if it's too fast or too slow you can tell me now or if there's anything that is not clear you can also think until the next time whether you want to ask me some questions so I think tomorrow we meet early no is that correct what at four ah although that's good because I'm actually not a very happy early riser um okay so tomorrow we meet at four and maybe I will try by tomorrow to have some some homeworks which then you have one week to do there should be some elementary things to check with these things so I mean I hope that you know that you find what I told you until now quite elementary and kind of easy maybe it's a bit faster than usual but it's things that you normally would already know and you know basically I've given you some simple definitions and I have given you straightforward applications mostly have a bit many definitions but you're already familiar with them you know with time obviously we it will get slightly more advanced and then you have to you know maybe make a bit more effort but in some sense we do start from zero and everything that is being used kind of is explained here anyway okay thank you for today