Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Permutation group G help

  1. May 17, 2010 #1
    i'm having trouble to show that if P: G1 --> G2 is a group homomorphism, then the image, P(G1) = {g belongs to G2 , s.t. there exists h belonging G1 , P(h) = g}, is a subgroup of G2


    Let G be a group, and Perm(G) be the permutation group of G. Show that the
    map Q : G --> Perm(G) g --> Qg (g is a subscript of Q) such that Qg(h) = gh (g is a subscript of Q) is well-defined, 1-1 and a group homomorphism, where g, h belong to G.

    Suppose that G = Z3 = {e, a, a^2}, a^3 = e. Labelling the points of Z3 as {1, 2, 3},
    with e = 1, a = 2 and a^2 = 3, give the permutations Qa and Qa^2 , explicitly. (a and a^2 are subscripts of Q)

    For the first part: Do i have to show the closure, identity, inverse and associativity

    For the 2nd part: How do i show that the map is well defined?

    For the third part: I'm not sure where to start?
    Last edited: May 17, 2010
  2. jcsd
  3. May 17, 2010 #2
    Consider the definition of a homomorphism itself, then the elements on the image should form a group(which is a subgroup of G2)
    Try applying the subgroup test to P(G1), whichever one you've learned.

    My group theory is kinda rusty, so I'll leave the rest to others.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook