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Prove that there are no simple groups of order 792 or 918 ....

Question

Prove that there are no simple groups of order 792 or 918 .

Prove that there are no simple groups of order 792 or 918 .



Answers

Proof Prove Property 2 of Theorem 9.17

So you want to prove that? Nine. The bar. Oh, man. Minus one. It was the musical bye meat. So by induction now the this case, right? He's kidding. Seven equals one. So that is nine. The bar one minus one. Well, that is eight. Jesus visited by my hate for this case is done for the inductive step. Um, assume ah, the nine to the online is one minus one is the musical, but it But it sort of means that these numbers will tow a times. So men did your, um So you ever been trying to the end one The physical too. Nine a the n minus one, minus eight. I say you have these two cars. So on these these miners thing sort of this company factor as line. Thanks. 90. And Linda's one, Because I know, uh, yeah. Musical One last name my was one, uh, minus one plus eight. But by the inductive apart. This is We know that these numbers of although it is a him I sort of got that number is nine. That's them. I'm state for these numbers. My sake, which is clearly visible. Way it. He's be civil by eight for these, Uh, the sends, uh, you have been Dr Step on this case by induction. This is different. He's Drew for any sure or yeah, me and yeah.

Okay, We've got eight to the end, minus three to the end, and I say that that's divisible by five. Okay, um, we want to make sure that this work so that we can even dio induction with it. One way that we can do that is just simply by plugging on one. I'm so we'll try that out. Eight to the first months. Three to the first. That does give us 55 is divisible by five. We'll assume that K is true, so that if a to the K minus three to the K will assume that that does divide by five. So let's see if this works for the K plus one the next into induced step. All right, so we'll have eight to the K plus one minus three to the K plus one. Now, as you'll probably get used to an induction, we can always rewrite that exponents as eight to the cave. Times eight. We'll put a one there just to remind us what we did on that step minus three to the K times. Three to the first. Okay, um, we're interested in five, so let's see if we can rewrite things in terms of fives. I'm gonna try and rewrite the eight right Love eight to the cave. And that's just five plus three. Right? Then we'll have three to the K times three and that that is going to help us, right? We've got a three in both of these terms, but it's do one step at a time here. We re write this as five times eight to the cape, plus three times a to the K no right minus three times three to the K. Um, put these in parentheses. So a bit more proper. Uh, we'll have five times eight to the K. That's of course, divisible by five. Since we got a five right there five times, anything is divisible by five. On the second piece, we have a three that we can maneuver out. We'll have eight to the K minus three to the K in parentheses. Then this. You may recognize right here as P F. K. That is divisible by five. So that's divisible by five. That means anything you know, any number we can call the P. Times five is divisible by five. Same idea here. So both of these are divisible by five. If you add two numbers that are divisible by five, they are still divisible by five. Okay, one further step. Maybe just to to write this home a little bit. And I just said that adding two things divisible by five is divisible by five. But we can see this. Let's take the example. I have the peas in here still five and p times five. If we were then Teoh, we can call one of them baby. Q. Right. If we had five, please plus five. Q. You could factor out of five, right, and you'd have five times People's que than this one term, right? You can see it more easily now that that would be divisible by five.

Okay. Snacks One says that three to the to end minus one is divisible by eight. Uh, okay, for all natural numbers end. So if we're gonna do induction, we need to make sure that n equals one. Works. So three times two to the first power by this one. That's nine months. One that is eight. And that is divisible by eight. So that works. Let's assume that K is true. So 3 to 2 K minus one is divisible by eight. And then we'll work on the cape us one piece. All right, so we have three times to times K plus one minus one. Let's go ahead and distribute the two and the exponents miscued this over lips. Let me screw this over a bit. So that room two K plus two minus one. All right, I'll rewrite. That is 3 to 2 K. Times three squared, OK, minus one. And so that's that's kind of nice. Already as one step, I'm gonna go ahead and maybe see if I can write this as something else. Keep in mind, I'm trying to get these kind of things in here. If I can get three to the two K minus one as a term on its own. That will help me. So I can. Well, I just have a nine right here right now, But I wanted to I could destroyed it as nine times. 3 to 2. K minus one. Uh, I could also rewrite nine as a plus one if I wanted to. Right? That's the same thing. And then I have eight times three to the two K. That's divisible by eight. Right. I also have, uh, 3 to 2. Kate minus one. All right, so this one the induction we saw right here, that is divisible by eight. Okay, so that one works, and we just said this is divisible by eight. All right? Now, uh, you still kind of same step that I often skip over. Um, we have to prove that two numbers that are divisible by eight are still divisible by eight. So you can just turn this one temporarily as p. And this one is cute. OK, que being this whole entire thing. Right? So we'd really What we have here is eight, um, times p right. And then eight times one technically, right now, because any this Q is always divisible by eight. All right, so we're adding these two things together, some kind of number divisible by eight. We'll call it a two. All right? We can factor that eight out of both of these. And you can see this combined thing is also divisible by eight.

Well, The open interval 01 injects into the closing tomorrow, 01 But just taking X. There goes to the very same number and this is injected. So the carnality off the open set well, interval is lower equal than the carnality off the clothes set now for the other way around. That's the final map that goes from the close set 01 to the open side, 01 where every X between 01 included. He's mop to, let's say, one house plus ex quarters. These graphically means that the numbers from 0 to 1, our scent too well when x zero just goes to 1/2 and when X is one, we go 1/2 plus 1/4. That's 3/4 and this little interrupt from one after 3/4 is inside the open in turmoil. Well, here we have zero. It goes all the way to one off and then here because all the way to one and of course, is my prison detective as well. And so we know that the current director of the close said 01 is lower equal than the car limit of the open, said 01 and therefore the carnality off these two sets of thes two intervals must be the same


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