1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

  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


  4. Mar 5, 2009 #3
    Yes, I don't need to say "Assume", Thanks!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook