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Help with permutation groups

  1. May 14, 2010 #1
    Help with permutation groups....

    How do i 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 to G1 , P(h) = g}, is a subgroup of G2

    Also if we let G be a group, and Perm(G) be the permutation group of G. How do i show that the map

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

    Now suppose that G = Z3 (Z subscript 3) = {e, a, a^2}, a^3 = e. If we Label the points of Z3 as {1, 2, 3}, with e = 1, a = 2 and a^2 = 3, how to we give the permutations Qa and Qa^2 , explicitly. (where again a and a^2 are subscripts of Q)
     
  2. jcsd
  3. May 14, 2010 #2

    jbunniii

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    Re: Help with permutation groups....

    For the first part, assume you have two elements [itex]a, b \in P(G_1)[/itex].

    Thus [itex]a = P(x)[/itex] and [itex]b = P(y)[/itex] for some [itex]x,y \in G_1[/itex].

    What condition(s) must these elements satisfy in order for [itex]P(G_1)[/itex] to be a subgroup of [itex]G_2[/itex]?
     
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