1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    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 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

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    User Avatar
    Staff Emeritus
    Science Advisor

    (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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Abelian Groups
  1. Abelian group (Replies: 9)

  2. Abelian group (Replies: 1)

  3. Abelian Groups (Replies: 9)

  4. Abelian group (Replies: 2)

Loading...