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Group action

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Let X be a G-set and let x and y denoted elements of X.
    a) If x in X and b in G, show that S(bx) = bS(x)b-1
    b) If S(x) and S(y) are conjugate subgroups, show that |Gx| = |Gy|


    3. The attempt at a solution
    Let S(x) = {a in G | ax=x}
    Let S(bx) = {a in G | abx=x} => abx = x => bx = x => bxb-1 = xb-1

    bS(x)b-1 = baxb-1 = bxb-1 = xb-1

    Thus S(bx) = bS(x)b-1

    b) Since S(x) and S(y) are conjugate subgroups, |S(x)| = |S(y)|
    so by |Gx|=|G:S(x)|, |Gx|=|Gy|
    correct???
     
  2. jcsd
  3. May 6, 2009 #2

    matt grime

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    Science Advisor
    Homework Helper

    No, that does not hold. That abx=x does not imply that bx=x, as that would mean that b stabilizes x which is certainly not (necessarily) true.



    Whoa, there. You've now equated a group with an element in a group.

    Start again with b^-1abx = x and see what you can say.
     
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