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Group theory question

  1. Dec 20, 2007 #1


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    1. The problem statement, all variables and given/known data
    G is abelian, A is normal in G, B is a subgroup, a1, a2 in A, b1,b2 in B, c_g denotes the congugation by g automorphism. why must

    [tex]a_1c_{b_1}(a_2) = a_2c_{b_2}(a_1)[/tex]

    imply that [tex]c_{b_1}(a_2)=a_2[/tex] and [tex]c_{b_2}(a_1)=a_1[/tex]

    3. The attempt at a solution

    In other words, why couldn't there exists a, a' in A such that [tex]c_{b_1}(a_2)=a[/tex] and [tex]c_{b_2}(a_1)=a'[/tex] and [tex]a_1a=a_2a'[/tex]??
  2. jcsd
  3. Dec 20, 2007 #2


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    Conjugation is trivial in an abelian group: ghg-1 = gg-1h = h.
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