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Math Help - hard questions in groups - please help me!

  1. #1
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    hard questions in groups - please help me!

    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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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by themanandthe View Post
    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.
    Using a simple application of Sylow's theorems, you want to show that there is only one Sylow 11-subgroup. Why is this sufficient?

    Quote Originally Posted by themanandthe View Post
    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.
    For (a) I ask must ask you a question...what does an element of order 3 in A_5 look like (in cycle notation)? Can you count how many elements have this form?

    For (b), you know that there are ` n' elements of order 3. Each of these must appear in some Sylow 3-subgroup of order 3, and so they must appear in pairs in this subgroup. So your answer is n/2 (do you understand why this is)?

    You should check that this answer is valid by plugging in Sylow's Theorems (they do not give you the actual answer in this case, just a couple of possible answers).

    For (c), what is the order of a Sylow 2-subgroup? I shall call this number ` m'. Now, can you think of a well-known result about groups of order `m'? For the next bit, every group of order m is a Sylow 2-subgroup (why?), and as all such groups are conjugate they are all isomorphic. Therefore, to finish the question you need to find a subgroup (any subgroup!) of A_5 of order ` m'. What does this group look like?

    Quote Originally Posted by themanandthe View Post
    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).
    Hmm...(a) and (b) are actually false, but I suspect you are only looking at finite groups and so are disproved by simple order arguments. (For (b), a counter-example is the integers...I can't think of a counter-example for (a) off-hand EDIT: Take the infinte cross-product of \mathbb{Z} with itself. This is an infinitely generated abelian group. Then, quotient out one of the \mathbb{Z}'s...).

    As a hint for (c), `Cayley's Theorem'.

    For (d), I believe your first counter-example comes at order 6. However, you should note that S_i \leq S_j for i \leq j. So...

    And I will leave you to solve (e). You proof should start `Let g \not\in Z(G)...' (again, it is false).
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  3. #3
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    Swlabr - wow! thanks a lot for your help!!! really!

    i'm kindda new at this so i'm trying to understand what you have wrote but still, i would really appreciate if you could simplfy a bit more for me..

    1. Using a simple application of Sylow's theorems - how do i use it?
    2. (a) - i really dont know.. sorry about that.
    2.(b) - do you understand why this is)? now i dont..
    2.(c) Therefore, to finish the question you need to find a subgroup (any subgroup!) of of order `'. What does this group look like? can u explain? i can't get it right..

    3.(a) Then, quotient out one of the 's - how exactly i do that?
    3.(c) - Cayley's Theorem' -???? i can't see it..
    3(e) - sorry but again i dont understand how..

    thanks a lot! sorry for all the question. just starting to learn this subject..
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  4. #4
    MHF Contributor Swlabr's Avatar
    Joined
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    Quote Originally Posted by themanandthe View Post
    Swlabr - wow! thanks a lot for your help!!! really!

    i'm kindda new at this so i'm trying to understand what you have wrote but still, i would really appreciate if you could simplfy a bit more for me..

    1. Using a simple application of Sylow's theorems - how do i use it?
    Have you covered Sylow's Theorems? If so, go back to where it was covered. There should be a similar example.

    Essentially, Sylow's Theorems are three theorems.

    -The first says `if |G| = p^nm and gcd(p, m)=1 with p prime then there exists a subgroup of order p^n in G'. Such a subgroup is called a Sylow p-subgroup.

    -The second says that all Sylow p-subgroups are conjugate for a fixed p. Thus, they are always isomorphic, and if there is only one it is normal.

    -The third gives some information on the number of Sylow p-subgroups for a fixed p, denoted n_p. Specifically,

    n_p | m
    n_p \equiv 1 \text{ mod }p
    and a third condition to do with the normaliser.

    You want to use the first two of those conditions to find the number of Sylow 11-subgroups. What does this mean? How can you apply this to your question (you will need to use the fact that the center of a p-group is non-trivial for p prime.)

    Quote Originally Posted by themanandthe View Post
    2. (a) - i really dont know.. sorry about that.
    2.(b) - do you understand why this is)? now i dont..
    2.(c) Therefore, to finish the question you need to find a subgroup (any subgroup!) of of order `'. What does this group look like? can u explain? i can't get it right..
    Okay, first things first. What does an element of order 3 in S_5 look like? Which of these are in A_5?

    Quote Originally Posted by themanandthe View Post

    3.(a) Then, quotient out one of the 's - how exactly i do that?
    3.(c) - Cayley's Theorem' -???? i can't see it..
    3(e) - sorry but again i dont understand how..

    thanks a lot! sorry for all the question. just starting to learn this subject..
    Ignore what I said for 3(a) - just use counting arguments to look at the finite case.

    For 3(c), do you know what Cayley's Theorem says? How many generators does S_n have?

    For 3(e), you have an element outwith the center. So, you want to put this in the center of a subgroup....I will, however, leave you to ponder/solve it.
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