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Homework Help: Abelian Groups

  1. Dec 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Let (G, °) be a group such that the mapping f from G into G defined by f(a) = a^(-1) is a homomorphism. Show that (G, °) is abelian.


    3. The attempt at a solution

    f(a) = a^(-1)
    f(a^(-1)) = f(a)^(-1) = (a^-1)^-1 = a

    in order for a group to be abelian it needs to meet the requirement a(i)*a(j) = a(j) * a(i)
    ° 1 a a^-1
    1 1 a a^-1
    a a a^2 1
    a^-1 a^-1 1 a

    since each side of the diagonal are the same then (G, °) is abelian.
     
  2. jcsd
  3. Dec 16, 2008 #2

    HallsofIvy

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    You are assuming that G has only three members? Of course, every group containing 3 members is isomorphic to the rotatation group of a triangle which is abelian.

    What if G contained six or more members?
     
  4. Dec 16, 2008 #3
    I realize that my answer only takes into account for 3 members but I am having trouble coming up with a solution for all possible amounts of members
     
  5. Dec 16, 2008 #4

    HallsofIvy

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    (ab)-1= a-1b-1 if and only if a and b commute.
     
  6. Dec 16, 2008 #5
    Interestingly enough, such a homomorphism must be an automorphism...not that I think it helps for this problem. HallsOfIvy is dead on with his hint.
     
  7. Dec 16, 2008 #6
    So to follow up on that hint, see where f maps a, b and ab
     
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