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Abelian and homomorphism

  1. Mar 5, 2009 #1
    1. The problem statement, all variables and given/known data
    If G is any group, define f:G -> G by f(g) = g^-1
    show that G is abelian if and only if f is a homomorphism.

    3. The attempt at a solution
    Suppose G is abelian.
    Let a,b in G.
    f(ab) = (ab)^-1 = b^-1 a^-1. Since G is abelian, b^-1 a^-1 = a^-1 b^-1.
    we need to show that f(ab) = f(a)f(b)
    f(ab) = f(a)f(b) = a^-1 b^-1 by define f.
    so f is a homomorphism.

    Suppose f is a homomorphism.
    Let a,b in G.
    Since f is a homomorphism, f(ab) = f(a)f(b) = a^-1 b^-1.
    By define f, f(ab) = (ab)^-1 = b^-1 a^-1
    Assume that a^-1b^-1 = b^-1 a^-1
    a a^-1 b^-1 = a b^-1 a^-1
    b^-1 = a b^-1 a^-1
    b b^-1 = ba b^-1 a^-1
    e = ba b^-1 a^-1
    a = ba b^-1 a^-1 a
    ab = ba b^-1 b
    ab=ba, so G is abelian.

    Correct?
     
  2. jcsd
  3. Mar 5, 2009 #2


    It looks good to me. Besides,here when you say Assume that a^-1b^-1 = b^-1 a^-1 you don't really need to say so, because this follows imediately by assuming that f is homomorphism.

    For the first part, it is correct, however i would write it this way

    f(ab)=(ab)^-1=b^-1a^-1=a^-1b^-1=f(a)f(b).

    Cheers!
     
  4. Mar 5, 2009 #3
    Yes, I don't need to say "Assume", Thanks!
     
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