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Additional Group Theory Issues

  1. Mar 16, 2009 #1

    ZTV

    User Avatar

    I really don't get this group theory stuff at all. These should be simple questions, but alas not...

    1. The problem statement, all variables and given/known data

    Assume that * is an associative operation on S and that a is an element of S.

    Let C(a) = {x: x is an element of S and a*x = x*a}

    Prove that C(a) is closed with respect to *

    2. Relevant equations

    Unsure

    3. The attempt at a solution

    To be perfectly honest, I don't understand the notation or anything. I have no clue where to start. I know that to prove closure I have to show that when g, h are elements of G that g*h is also an element of G. Does this mean I have to show that a*x is an element of S?

    ~~~~

    1. The problem statement, all variables and given/known data

    Prove that Aut(Z3) is isomorphic to Z2 [Z3 and Z2 the group of modulo classes, eg. Z3: { [0][1][2]}

    2. Relevant equations

    Once I've found Aut(Z3) which I think I've done, I need to show that Aut(Z3) is one-to-one, onto and operation preserving (homorphic)

    I found Aut(Z3) to be {[0][1][2] , [0][2][1]}

    3. The attempt at a solution

    With Aut(Z3) = {[0][1][2] , [0][2][1]}
    and
    Z2 = {[0][1]}

    I can see how they are supposed to be isomorphic already.

    I consider f: {Aut(Z3)) -> Z2}

    I'm thinking that showing that they are one-to-one and onto may be trivial, but I'm not sure.

    I also looked at f ([0][1][2]) = f ([0][2][1]) and proving [0]=[1] for one-to-one but this doesn't make sense because [0]=/=[1] ... hmmm... perhaps I'm being silly.
     
  2. jcsd
  3. Mar 16, 2009 #2
    For the first question take two elements of the C(a), say x and y, and show that x*y is in C(a), i.e. (x*y)*a=a*(x*y)

    Second question, you need to find the Automorphisms on Z3 which you have. Aut is a group under composition. Label the two automorphisms you have and make an isomorphism (which is pretty obvious) to Z2. The problem you are having is that you got the wrong impression what Aut is. It is a set of functions, so Aut(Z3) = {[0][1][2] , [0][2][1]} is not true. It consists of two functions, one of which is the identity, and the other one swaps [1] and [2].
     
  4. Mar 16, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

     
  5. Mar 17, 2009 #4

    ZTV

    User Avatar

    So what are the elements of Aut(Z3) then?

    Functions f that map Z3->Z3 such that fZ3 = Z3 ?

    If this is the case...

    I now have aut(Z3) = {fa fb} and Z2 = {[0][1]}

    I'm trying to show that they are homomorpic/operation preserving.

    I have to define a function y that maps
    y: Aut(Z3) -> Z2 is this correct? If so, how do I show it is operation preserving?

    Once I have that it is homomorphc, the fact that it is one-to-one and onto is trivial because fa -> [0] if it is homomorphic, so fb must go to [1] hence one-to-one and onto. Is this correct also?

    Thanks
     
  6. Mar 17, 2009 #5
    Yes, mind you fa is the identity map. If you have shown it to be a homomorphism then it preserves the operation (which is what homomorphisms are). The bijection is clear, homomorphism shouldn't be too hard to prove.

    As a general rule of thumb you want to specify how y maps, its the map as you said that takes fa to [0] and fb to [1].

    Think of homomorphism as structure preserving, if f is a homomorphism then f(ab)=f(a)f(b), so they practically have the same operation. Isomorphism tells you that you just essentially relabeled your elements, because it is bijective and operation preserving.
     
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