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August 22nd, 2010, 11:23 AM   #1
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question in groups

hi! I really would appreciate if someone can solve these questions..
please.
thanks a lot!


1. let G be a group from order 297. prove that there is a subgroup K of G wich
the center of G/K is not Trivial.

2. look at A5 :
a. how many numbers from order 3 are in A5?
b. how many 3-Sylow groups are in A5?
c. let H be 2-Sylow subgroup of A5. prove that H abelian and find wich abelian group
H is isomorphic.

3. prove or disprove:
a.there is no group G and a normal subgroup H of G (when H is not G or {1G}) so that G/H isomorphic to G.
b. there is no group G and subgroup H [so that H is not G] so that G
isomorphic to H.
c. for all n natural there is a group G and a subgroup H
[wich both are Independent in n] so that rank(G)=2 and rank(H)=n
d. for all G group from order n, there are no m<n (when m belongs to N)
so that there is monomorphism G->Sm.
e. for all group G , H subgroup of G so that: Z(H) is a subgroup of Z(G).
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August 22nd, 2010, 12:36 PM   #2
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Re: question in groups

Hello themanandthe.

That's quite a list of problems. Maybe you care to post the work you have done so far?

Best,

Ormkärr
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August 22nd, 2010, 02:57 PM   #3
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Re: question in groups

i know it's a lot. sorry about that, but i dont know how to solve them.

again, thanks for the help
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August 22nd, 2010, 06:23 PM   #4
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Re: question in groups

Hi again themanandthe.

I wasn't kidding---if you just post some of your work on the problems, it'll be more likely that you'll get someone to help you out. But presented as a long list of problems with the demand that they be solved, you're likely to put some good posters off from responding. After all, there's a bit of thinking that needs to be done here, and for it to be done pro bono, people are more prone to submit their help if the request is accompanied by some (perhaps false) modesty, or at least the appearance of an honest attempt.

It is as much a matter of diplomacy to petition help as it is an appeal to pity, or the presentation of an opportunity for another to stroke his ego, or any other thing. Learn this simple truth, and you will find as much help as you need.

Best of luck with your queries,

Ormkärr
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August 22nd, 2010, 07:59 PM   #5
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Re: question in groups

hey again..
wow! ok first im very sorry that you look at things like that. that is not what i meant.
you said:
"with the demand that they be solved" - i think i asked not demanded sorry if it seems like that..
"if the request is accompanied by some (perhaps false) modesty, or at least the appearance of an honest attempt". - i dont know why you said that. i really (with all the honesty i have got) dont have any clue of how to solve those question. therefore, i asked the help of the forum. i dont know even how to start.
anyways, if anyone would like to help me it will really help.
if not, thanks anyway.
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August 22nd, 2010, 10:16 PM   #6
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Re: question in groups

Hello themanandthe.

I'm sorry, I wasn't meaning to criticize you personally so harshly, I really hope you were not hurt by my words. I simply wanted to let you in on how I've noticed things go around here during my couple months of membership.

I wasn't necessarily implying you haven't done any work, but if you are taking abstract algebra, or are asking the questions you posed, you must obviously have some idea of the objects of inquiry, so maybe---in order to get replies---state what you know about, say, the quotient group, or Sylow p-groups, or the rank of a group, etc. (whether it helps with the solution or not) so that a starting point is established. Otherwise, I am sorry to say, it frankly doesn't bode well for getting a ton of replies. For your sake, I hope that is not the case, and that you get all your questions answered.

As far as the questions you pose are concerned, if I were you I would begin with the third question, since the other two will involve specific sorts of arguments, as the group (or at least its order) is given. The third tests your ability to come up with counterexamples, or prove a proposition in a more general setting, so in my opinion it's a good warm-up.

Take 3.a. for instance. One natural place to start is with the First Isomorphism Theorem, or even just facts about the quotient group. If an isomorphism ? between G and G' is given, then ker ? is trivial (why?). But every normal subgroup H is the kernel of a homomorphism, namely, the canonical projection ? : G ? G/H. So, if G ? G/H, then the kernel of the canonical projection is trivial, and hence, H is trivial. But this contradicts your assumption.

For 3.b., we are only dealing with an isomorphism between a group and its subgroup. If G is finite, what prevents the isomorphism from arising? [Hint: Look at the orders!] Therefore, a counterexample must involve an infinite group. So look at the simplest infinite group you know, and see if you can think of a nice, even-tempered counterexample. Oddly, it's not as hard as it might sound.

Anyhow, I will have to call it quits for now.

Best of luck with your questions.
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August 22nd, 2010, 11:59 PM   #7
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Re: question in groups

Greetings themanandthe,

I have some general advice (which hopefully many will read) and some (hopefully helpful) comments on your problems.

Stylistically, when asking for help, one should never ask for full answers. It is much more interesting and fruitful to help someone with specific questions or general big-picture concerns than to do their homework for them. As a student, you won't learn from help unless you have spent time working on and thinking about a question; reading a solution is no more helpful (and often less helpful) than reading what's already in a book, as far as actually learning and understanding a subject is concerned. If you haven't actively thought about a problem, it's very likely that a subtly in an explanation may escape you. At best, we can give you bricks and give advice as to how to lay them down; you must provide the mortar and put everything together. But if you haven't thought about the problem, it will be far less clear how everything should fit together, so you'll make most of your building's wall and the final corner won't meet up because you've just been laying down the bricks without really understanding what goes where.

So on the one hand, I urge everyone to spend AT LEAST half of an hour (the more time, the better) trying to solve a problem before asking for help from anyone, be it online or elsewhere. If you have no idea where to start on a question for a class or out of a book, start by looking at all of the theorems that appear to be even tangentially related to the question which has been covered so far. Try to connect the theorems and the definitions in ways that seem to go in the direction of the problem. If you don't understand a theorem or a definition which you believe to be relevant, then you need to spend time trying to identify what it is you don't understand and try to figure it out. If you have trouble figuring out a confusing detail in a proof or statement of a theorem or a definition, that is an excellent question to bring to other people as they may be able to clarify what you don't understand and give you a valuable perspective you may not have had. The better you understand the statements and proofs of theorems and the better you understand the definitions of the objects you are considering, the easier everything is.

If you feel that you have a generally good grasp of the material presented in the class or in the book relevant to the problem, then you should be set up to make an effort to solve the problem, so spend at least a half hour trying put things together in any way you can think of. Once you've spent some time considering a problem, then go ask someone for help. Be sure to tell them what the problem is, where you have gotten, and what else you have tried. If you've thought about a problem for at least a half hour, you should have something to say. Even if all of your ideas seem bad or foolish in retrospect, telling someone what you have tried is very helpful. It tells them how you are thinking about the problem so they can correct your misconceptions or give you the little nudge you need to be able to fit the pieces together more nicely so that you can get further. It also tells them what you know, so they don't try to provide help from a prospective that you have never seen.

I would also urge people to fully disclose the nature of the problem: if you are stuck on some problems out of your first abstract algebra set (are the problems from a book, if so which one, or just given by a professor), if they are related to a research task, if they are problems off the internet, if you just made them up, or whatever the case may be. Personally, I like helping people learn and the more information you include in your request, the more likely I feel that you take what you are doing seriously and that you actively want to learn, not just get answers. As Ormkärr was saying, the world is not short on problems to solve and the people who know how to help you won't learn much from doing three random questions; if they wanted to spend their time solving problems, they would go find problems that interested them. In general, it is your job to convince them that their time would be well spent trying to help you.


Now on to the math. What is special about 297? In finite group theory, the primes dividing the order tend to be important. In particular, we know . So what are the sizes of the Sylow-p subgroups? What do we know about how many Sylow-3 subgroups can there be? Sylow-11 subgroups? In particular, what do we need to even be able to consider quotient groups?

Let's start with the most basic question: what is ?

For the third problem, Ormkärr has given a nice place to start.
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