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D. Let F and Fz be distinct reflections in dihedral group Dnpoint ) Show that FiFz # Ro_ points) If FiFz = FFi, show that FFz = Risg_points) Prove that an Abelian g...

Question

D. Let F and Fz be distinct reflections in dihedral group Dnpoint ) Show that FiFz # Ro_ points) If FiFz = FFi, show that FFz = Risg_points) Prove that an Abelian group with two elements of order 2 must have subgroup of orderpoints) Find all subgroups and determine the subgroup lattice of Zaz 3 points) Suppose G is group and a,b € G. Suppose that the order of is odd, and aba-! =6-1 Show that b2

D. Let F and Fz be distinct reflections in dihedral group Dn point ) Show that FiFz # Ro_ points) If FiFz = FFi, show that FFz = Risg_ points) Prove that an Abelian group with two elements of order 2 must have subgroup of order points) Find all subgroups and determine the subgroup lattice of Zaz 3 points) Suppose G is group and a,b € G. Suppose that the order of is odd, and aba-! =6-1 Show that b2



Answers

Find the distance between each pair of points
with the given coordinates. Then, find the
midpoint of the segment that has endpoints
at the given coordinates.
11. (1.-6).(-3,-4)
12. (a,b). (a +3.b + 1)
13. Determine whether the points A(-5.-2).
B(3, 1), C(10,3), and D(2-3) are the
vertices of a parallelogram. Justify your
answer

All right first let's suppose end plus five is even and we'll show that three and plus one is even if N plus five is even, then N has to be ought. Therefore N has to equal to K plus one. For some integer K. Then you can compute three N plus one equals six K plus three plus one, which equals six K plus four. And simplifies to two times three K plus two. In other words it is two times in integer. That is why at three N plus one must be even. It equals two times the integer three K plus two. For the other direction three N plus one is even implies and plus five is even well actually show that by contra positive, we'll assume and plus five is odd. And showed that three and plus one is odd. Well, if and plus fine design then and is even so N is two times some integer K. So three N plus one equals six K plus one, which equals two times the integer three K plus one. Therefore three M plus one is odd. Now for part B. One of the key facts you need is that a vertex in the complete graph on advertises has degree and minus one. So you look at the complete graph on three N plus two virgins is K. Three N plus two. That graph has an oiler circuit. If and only if all those degrees of the verses which are all the same. It's going to have to be even so three N plus one. The degree of all those virtues is that's going to have to be even by part A. We've proven that three N plus one is even if not only if N plus five is even and what's and close fine. It's the degree of all the vertex is in cayenne plus six. And if that's even if N plus five is even, then that means that complete graph can plus six has an oiler circuit. Therefore, the chain of equivalence is uh yields that can K. Three and plus two has an oiler circuit. If and only if KM plus six has an oiler, Sir.

Okay for this question you're being asked which point is the closest to the Y. Z. Plane? And the answer is Point C. The next part of being asked what is the distance from the Y. Z plane to this point? And the distance is minus line? The next question is which point is the farthest away from the XY plane? The answer is .8. What is the distance from the xy plane to this point? And the answer is -17. Which point lies on the XZ plane? Around six point we. And this is the final solution to the question.

Hello. So these are the two diagrams. Uh So I don't think this diagram they are neither similar nor congruent. So for similar, you know, objects or figures. Uh huh. The relationship between them should be a school factor. Okay so for example we should live on this side on this side, that's this side. And that side you see that the school factor is too right? But look at the other side, this is 15.8 and she's 10.8. That's not true. It has to be constant throughout. Okay so and of course uh No they're not 10.8 and 19.8. You know it's not it's not it's not let's go fuck that is not true. If you look at this side too, it's the same thing. Uh Look at his blue and this blue that is also not a scale factor of two. So they are not similar. It took a congruent. They're supposed to have you know the same size, would you be the same size? And obviously the sizes are not the same. Okay. I would assume size and some angles. So do I need to say so They are kinda congruent. No similar. Okay.

In this question. First of all, it is being asked to find the distance between the points. The first pair of points is given to us is of one comma minus six and minus three comma minus four. So to find the distance, I'm applying the formula that is the root of X two minus x one Holy square plus while to minus y one hole is where this provides me with a lot of extra minus x one means minus three minus one hole is where plus y tu minus environments minus four plus +64 weeks. We're okay after solving it, we get here, I can say that the root of 20 and it can be written as to route five. Ok, this is the distance here. That is to route five. Now we have to calculate here. The middle point also. So I am finding the middle point also for this question, we know that the formula for the midpoint is excellent plus six, two by two, comma y one plus by two by two. Okay now I apply it here so I should get here one minus three by two comma minus six minus four by two. That provides me, I can say that minus one comma minus five. Okay now similar things we have to do for the second point. The second point given to us is I can say that that is number 12 years, which is a comma B and a plus three comma B plus one. Now we are applying the distance formula to find out the distance that is the root of X two. That is a plus three minus X one is worth plus why do that? Is we plus one minus y one. That is based where? Okay, this provides me after the cancellation of A. And B. We got here nine plus one. That is the road of 10. Okay, this is the value of distance here. That is the road of 10. Now we have to also find the midpoint here. So the midpoint is again given by some of the extraordinary. That is A plus A plus three, divide by two. And some of the white coordinates that are B plus B plus one by two. This provides you a plus three by two comma B plus one by two. Okay this is development point coordinate. Now in the third part we are provided good. Uh what it says of I can say that parallelogram and we have to check whether it is a parallelogram or not. So forward this is are provided to us that our A. B, C. And D. Value of ways I can say that minus five minus two. Value of bees, three on my 11th and value of seas 10 comma three and value of days two comma minus three. Okay now I apply the distance formula to find the side baby. That should be given by route of extra minus means three plus five. Holy square and 11. Okay 11 minus two. Sorry this is not 11 year. Okay I can say that I have here one only. This is one so one plus two one last two. Holy square. So this provides me root of 73. Now I calculate the side city by the distance formula. That is a vote of 10 minus two. Holy square plus three plus three. Holy square. Okay this provides me a root off. I can say that 100 means 10. Now we can see that both the sides are not equally and since they are positive side they must be equal here. And you can see that the Arabian city are not equal. So you can say that this is not a parallelogram. Okay, this is not a parallelogram because the opposite sides are not equal here in this question. Okay, thank you.


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