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6. (15 pts) Assume p and q are primes with p q and that G is group of order pq: Use the Sylow theorems to prove that G has a normal subgroup of order q. Prove that ...

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6. (15 pts) Assume p and q are primes with p q and that G is group of order pq: Use the Sylow theorems to prove that G has a normal subgroup of order q. Prove that if q # 1 mod p, then G also has a normal subgroup O1' order p. Prove that the intersection of a Sylow p-group and Sylow q-group must have order 1_

6. (15 pts) Assume p and q are primes with p q and that G is group of order pq: Use the Sylow theorems to prove that G has a normal subgroup of order q. Prove that if q # 1 mod p, then G also has a normal subgroup O1' order p. Prove that the intersection of a Sylow p-group and Sylow q-group must have order 1_



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If $14 p-23=6 p-7$ then $p$ equals:

To talk about this question of intra determine whether the points and P Q. R. Or the three points whether they are a linear or not. If they are colonial, then we have to find a parametric equation. The line which is passing through these points. If they are not colonial, then we have to find the area of the triangle peak. You are. So the best way to there are actually a couple of ways to recommend whether three points and three D. Space our colonial or not. One of the way is we find that direction ratios taking two points at the time. And we'll see if the direction direction ratios are in the same proportion or not. If they are and it is called linear and if not then they're not calling you. So if we talk about the direction ratios that let's say we talk about PQ. Uh then the direction ratios would be uh would be uh let's say will be proportional to will be proportional to uh will subtract. Well, well look, you minus P. So negative four -2 is Nicholas six. Likewise this is four. And this is to and if we talk about the direction ratios of let's say pr Then that will be in the proportion of six, then this will be negative for and this will be this will be sixth and this will be nearly four. And this will again be six. So clearly PQ and pr are not really in the same, uh not proportional to each other proportional. I mean that I should be able to right direction ratio of peak. you should be a constant K times direction ratio of pr but that is not possible in this case. So definitely they are not linear if they are not linear than uh So we'll say that they are not well we knew if they're not linear then we can find the area of the triangle which is formed by it. So for that we need to vectors. We we have already found those two vectors. One of the factor is PQ. So the PQ is gonna be -6. Nearly 6 4 and two 64 and two. And P R P R. It's gonna be six and -4 and six. To find to find area of the triangle peak. You are we are going to find the cross product of these two vectors. So we're going to say that India area is going to be half of half of the cross product of these two actors. So for the cross product we know the formula. So that's E G K. That's going to be nearly 64 and two. And that's going to be 6 -4 and six. So if we expand this, that's going to be 1/2 times. Absolutely. I mean, the absolute value of the monuments of It's gonna be 24 plus mhm -J times. This is gonna be negative 36 -12 plus K times. This is going to be 24 -24. So this is just like come out as 1/2 times. 24 plus eight is going to be 6 32 32 I and this is going to be 30 40 48 plus 48. Check And the magnitude of this is going to be square root of square root of 3, 2 square Plus 40 years squares. We've got to use calculator here Swimming Crab calculator and find 22 square plus 48 square and square root of this. And there should be divided by two. So the final value is coming as the area of this triangle is coming as 28 84 sqits. And this is rounded up to our do the simple basis. So this is the required area of the triangle peak. You are. Thank you.

And this problem we're talking about the various properties of relations on sets. Specifically in this case we're talking about a set on the rational numbers, which we denote with Q. So before we start we are talking about reflexive, transitive, symmetric and anti symmetric. If you're not comfortable with the definitions yet, I would look at those prior to watching this video or maybe during because I think that's going to help you understand where I'm getting this information. Now, let's just review what we're told. We're told that this relation is true if and only if X is greater than or equal to Y squared. So for reflexive, our relations are is not reflexive. So I'm going to prove this by counter example. Let's consider the 0.44 Well, 44 is definitely not in our relation because four is not greater than or equal to 16. So are is not reflexive for symmetric. Again, we can do this by a proof by counter example. Are is not symmetric. Let's let X equal three and Y equals six. Well three is not greater than or equal to 36 but three is greater than or equal to one squared. So are is not symmetric for anti symmetric. We can do this little bit more proof based art is anti symmetric, so if X is greater than or equal to Y squared, and if Y is greater than or equal to X squared, then we can deduce then X is greater than or equal to Y squared, which is greater than or equal to X to the fourth. How did we get X to the fourth? While we squared X squared. And this is true because let's say that X equals zero and X equals one. Then if X equals zero and Y equals zero and X equals one, we can deduce that one is greater than or equal to pardon me, Y squared, which is greater than or equal to one. So clearly why is one? And this holds. And then finally for transitive, we can see that our relation is transitive because of X is greater than or equal to Y squared. And why is greater than or equal to some arbitrary element in our set Z. Why is greater than or equal to Z squared? So we can deduce that X is greater than or equal to Y squared, which is greater than or equal to why? Which is greater than or equal to Z squared. So we can say our is transitive. So I hope with this problem helped you understand how we can prove the various properties of relation on a set and irrational numbers.

Mhm. Mhm. In this problem we know that three propositions P, Q and R are true. We want to know which of the following propositions are true for a is negative P. Or creation of P. O. Or parenthesis cube and our par PSP. And it's a parenthesis, A negation of Q or R. And in Farsi P implies negation of Q and R. Any party B is equivalent or if and only if Q or negation of our. So we start with are a. It is negation of P or Q and R. So in this case and each of the examples we are working here. The first thing we gotta do is solve with what is inside parenthesis. So parenthesis is way of organizing the information. So we need to solve first with these inside parentheses. So we got to know the value of Q and R. Knowing that you and are are both of us are both true and because this is and on uh connector we know that it is true and both are true. So it is the case here. So two are true implies that cube the proposition Q and R. He is true because both are true in this case the proposition and is too. Yeah. So knowing that this is true here. Yeah. It is connected with negative segregation of P. Using the connector or you know that if one of the parts of the connector or where the one of the proposition that are connected through the connector or is true, doesn't matter the value of the other one and the proposition or would be true. That is in this case, we know that you and are these propositions that branches. This is true. So it is sufficient that information to know that the whole proposition with the connector or will be also true. So this implies that location of P. Or two are is true. Yeah, because one of the parts that are connected through the connector or here is true this case. This one thing so far is true. Part B is P. And inside parenthesis negation of Q. Or are. So we have again, forenza says we got a soul that before anything else. So you get to know the value of negation of Q. Or R. Yeah. And because we have a connection or here we know that if one of the connect for both of the connector, propositions are connected through before our true the proposition, the whole position will be true. This case are is true, so are true implies that negation of cuba or are is true true. So the propositions high parenthesis is true. And now we have aunt here proposition the connector and and the connector. And is true when both are true. We're both propositions that are connected through it to it are true and this case B is true and we have proof here that the proposition is separate. This this is true. So the connection and between them is true. So we can see here the P true and negation of Q are true implies that three and negation of Q or R. It's true. So we have that R. P is true as well as as it was for a Alice support. See for tbs he implies. Yeah. Thank you, parenthesis, negation of Q. And our so again we have parenthesis. We gotta solve that first. Yeah. And here we have to proposition negation of Q. And are these ones too. And because Q is true is one is false because Q. It's true and the negation of the true value is false and vice versa. So in this case we have this is false, this is true. And this implies that the connection of these two proposition is an end is false because the end proposition is true only if both propositions that are connected with the and operator are true. So we have falls the second part of the implications. So we have P true. Yeah. Mhm. And negation of Q and R falls and we know that Mhm. If uh the hypothesis or the first part of the implication is true for the implication, the whole implication to be true. We've got to have also through the the second element of the proposition. So in this case we are false or true implies false is as proposition is false. Okay, Yeah, so uh party is not true is false. Yeah, the protest and finally party, we have P. If and only if inside parenthesis, Q. Or negation of our So we first solve what is inside parenthesis. So we have Q. Is true and negation of ours falls because our is true and the negation of true is false. So the connection using or the or operator here give us give us our true value sufficient that one of the two propositions be true for the whole proposition is in the connector or is true, so it is okay because we have Q. Is true even though I guess negation of our is false, so Q. Or negation of our is true. And now we remember that the double implication or even only if connector is true if both propositions are are connected with it are of the same value that is both are true or both are false. Then the if and only proposition is true. In this case we have to be true. And the second proposition in this case is one here. Q or negation of our true because both have the same value. So we can say that if and only if connection between them is true. Yeah, so this is true. No, So we have the party is true. So the only proposition of the four, given that it's not true, is part C, which is false. So the other three A, B and D. R. Truth.

Sorry about the video being sideways but hopefully you can see it if you can't let me know. So first we have to find the like terms so there's P. And P. Those are both multiplication. And then we have our regular integers. So we're going to want to get everything um on one side and everything on the other. So we're gonna want all of our variables on one side and all are integers on the other. So the first way we're going to start doing that is we're going to start moving things over. So we're gonna do the inverse operation here which is addition. So we're gonna add 23 when we do that minus 20 negative 23 plus 23 is equal to zero and then negative seven plus 23 is going to be and a subtraction problem. And it's going to equal out to a positive 16. So we're gonna put that 16 down here and then we're going to bring our six P. Down. And because this is positive the operation in between is going to be plus. So let's bring our six P. Down Equals right here. And then on the other side that's cancelled out. So we're going to bring down our 14 pete. So now we have to work on getting all of our variables to one side. So we're gonna take this six P. And move it on over. So six P. Or minus six P. That's gonna cancel each other out and everything we do one side we have to do to the other. So those cancel we're going to bring our 16 down to the bottom and there's nothing else over here so we can just write are equal sign and then 14 p minus six P. Is going to be an 18. So let's go ahead and divide out here. So we're going to the opposite of multiplication, which is division, divide that by eight, eight x 8, cancelling each other out. So we're left with P equals 16 divided by eight, which is to thank you so much.


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