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3. Let Dn denote the dihedral group for regular n-gon with n > 3. Show that Dr has semidirect product structure_ Dn # Cn X+ Cz- What is ( Cz - Aut(Cn) in this ca...

Question

3. Let Dn denote the dihedral group for regular n-gon with n > 3. Show that Dr has semidirect product structure_ Dn # Cn X+ Cz- What is ( Cz - Aut(Cn) in this case?

3. Let Dn denote the dihedral group for regular n-gon with n > 3. Show that Dr has semidirect product structure_ Dn # Cn X+ Cz- What is ( Cz - Aut(Cn) in this case?



Answers

Give a complete mechanism for the mono-bromination of
pentane at the 3 position in the presence of light.
Name the product(s).

In this exercise, we're asked to find all subgroups of Paris to find all subgroups of the group Z nine. And of the secret groups 18 to do that. It's important to remember one theory. All subgroups of a secret group are sickly. That's one. And the order of the subgroups our divers are defenders of the earth as a group and there is only one supper for each division. So let's start with Z nine. Z nine is a secular group and The only dessert of nine is 3. There was the only difference of 9, 1 and three. Therefore there are subgroups A further nine and 3 Of Under 1. 3. Yeah. Uh huh. Moreover, they're both cyclic and there is only one sub group for each user. Therefore There's a lot 49 look like reunited. It's the next supper Uh says the nine. It seems that the nine is generated by X. The next separable T is a Murphy to 33. And it will be generated by excuse. And last one is a trivial group generated by the identity. Oh, now let's move on to Z to finance all subgroups of CNT. The users are one to divide it. Three divides 18, 16 points 18, 1918. And that's all. Oh, therefore there will be some groups of others 12369 and one of the one subgroup for charter and they're all going to be sick. So first we start with Z 18 And assumed the 18th generated by X. So from C 18 we will have to suburbs. One generated by that's cute. That's going to be the suburb of order six and the other generated by X squared is going to be subgroup of all your life. Then they both have the six. We'll have. Is he too is a suburb generated by X. Does that mean? And Z three generated by extra six and jeanine will also contain the three is sub because three D. White snake, but it won't contain Z. two is a suburb because two does not divide And both the two and the three will contain the identity group. This up.

Hi everyone Vinod they made so compounds, mm hmm, compounds. Our alcohol compounds, yeah, compounds consisting of consisting of more than one older than one Kyrill center. Mhm center in their structure. Uh huh structure. Now the structure of it should be structural missile to told. Yeah. Deborah mo your detainees and as follows. Okay, as follows. So This is CH three. Mm. Ch three. This is BR VR may so totally die brahma Butin. It's optically in Act two. It's um typically sure, inactive since sense it is support imposed and both on its mirror images. Merah images. Thanks a lot.

So let's start with the question. The question says prove that. Are said 10 elements has any two in minus one and two. And managed to buy six subsets containing exactly three elements whenever and is an integer greater than or equal to three. Okay, so what we can do is we can use the mic to prove this thing. Okay, we're probably using induction. So let's start by writing model things we have been given. Uh so it has given that N is an integer bit uh where N is greater than or equal to three. Okay. And to prove uh yes, I set with elements has and into n minus one into n minus two by six subjects containing exactly three elements. This is what you have to proof. Okay, so we are going to prove it by induction. Okay, Okay. So let us assume the first step let PN Sorry, uh let B n B a set bet. Okay. Elements. This is what was the question? Okay, and elements uh has N into n minus one in two in minus two by six subsets uh containing. Okay, Okay. Uh exactly three elements. Okay, first thing we have to write it in your question from you. Okay you have a new give the solution. You have to write this thing as it is. Uh Now let's do the basic step. Okay, so normally be a substitute uh be of one, but here it is given that N is less than equal to three. So basically you'll find B of uh three. That is N equals three. Okay, so was set with three elements. Uh Mintz is of uh Okay. Abc Okay. Uh uh The only subset um I just with three elements. Uh Sorry about the spelling mistake. Uh Three elements is one. That's a little Okay. Uh Okay, so there is only one subset with three elements. Okay, now let's see uh substitute in the formula and Legacy. Okay, so it was in in two in minus one in 20 minutes to buy six and 33 into three minus one is equal to 22 three minutes to within one. And this thing is six by six is nothing but one. Okay, so pay of three is true. Okay, now let's go to the induction step where we assume that P K is true. Okay, so if PK is true then uh what if a set with K elements has? How many elements came to came? I just one sorry, not elements. Then became this one came in this two by six subsets with exactly three elements. Three elements. Okay, we assume this to be true. Okay, I'm gonna take it as a question when this thing is true, PK is true. Okay, so what we need to do is we need to approve P k plus one is also true. Okay we prove this then we can say this induction is right. Okay so let's stand. Okay so you are set with K plus one element. Okay so how it will be? It will be in the form of heaven to uh Kay okay okay. Element and one more element will be there. Ok Plus one. Okay so uh you know the previous key element? Uh it's subset. Okay we know that the earlier subset. Okay uh even a two up to aK Okay it has uh K into k minus one and two K minus two by six uh subsets. Okay uh this from the previous thing we know this set has this many subset and it contains exactly uh three elements. Okay uh while uh okay in two k minus one by two subsets uh with exactly three elements, Three of which one will be but a k plus one. Okay. See then only it will be accommodated in this set. Okay? So what we can do is okay in two k minus one by minus two by six plus this subset. Okay, this number of subsets for this case came to k minus one by two. Okay, we will solve this. Okay uh Okay let's go to that. So what we can do is we can take came to k minus one comment from this. Okay, so this will be further, it is to k minus two by six plus one by two. Okay, so uh let's take a simple wheel. Okay minus one. Okay. Uh So it comes out to be uh okay minus two plus three. Okay, so this is came to k minus one and two K plus one by six and uh copy this expression to the next page. Uh Got a paste it. So how can I make it on the farm orphanages? Uh Let me try not in Formula. Just I have to bring the question in form of this. K plus one. Uh K plus one minus one. And K plus one minus to this farm. I have to bring Okay by six. So what I'll do is uh from the previous thing I had this thing. Okay I'll take this K plus one out. Okay, K plus one. I rearranged the terms. Okay And k minus one by six. Okay. Plus one is outside. Ok I can like this writing desk. A plus one minus one. Okay. Can like to take this? Okay. And the camera is one. I can like the desk A plus one minus two. Okay so I hope uh we got the terms that we required. Okay. Plus one plus one. Plus one. Okay, So from this, PK plus one is also true. So since PNS true, be cares true, B K plus one is also true. And the conclusion is by principle of mathematical induction, P N. Is true for all positive in teachers. And I hope you understand the problem. Uh, thank

Question. Give a complete mechanism for the combination of painting three positions in the presence of light the end products. So we are going to discuss one painting is reacting with the regulation, then what compounds we can get? So painting his vote Cst CS two CS 2 CS spruced. Yesterday it is reactive and that we are actually we are to catch new is producing we are ready to be our doctor. This we are not will attack on the two degree carbon, two degree carbon. So what we happen this will be uh you I'm alive, I'm alive like this. Then we will get to more of the br this be our dog. Will abstract. All right. We'll have tripped one edge plus founder. This one, correct as we are will be farm. And this really attack on this feeder on this field because they're waiting for. So see yesterday CS school you see it to be out. See enthusiastically. Right? So, I hope you are getting correct answer. Okay. Thank you. Mhm Yeah.


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