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Application in permutations group

  1. Nov 3, 2015 #1
    Hello
    I am studying for my exam and there's a question that i don't know how to solve, I have some difficulties with symmetric/permutations groups

    1. The problem statement, all variables and given/known data

    Consider a finite group of order > 2.
    We write Aut(G) for the group of automorphisms of G and Sg for the permutations group of the set G.
    Consider a1 and a2 ∈ Aut(G) two automorphisms.
    We suppose that the g=1 is the only element g ∈ G such as ∀ x ∈ G we have a1(g)*x=x*a2(g)
    For g,x ∈ G we have
    fg(x):=a1(g)*x*a2(g-1)
    Show that for each g ∈ G the application fg: G→G is in Sg

    2. Relevant equations


    3. The attempt at a solution
    I really dont know how to do that..
     
  2. jcsd
  3. Nov 4, 2015 #2

    fresh_42

    Staff: Mentor

    I don't get the problem here. Since all f_g are well-defined functions on the finite set G with an obvious inverse
    h_g (x) := a1(g^-1)*x*a2(g) = f_g^-1 (x), i.e. a bijection, they have to be in Sg, if Sg is Sym(G) the set of all bijections on G.

    The additional condition says that f_g = id(G) implies g = e. It is also true that f_g * f_h = f_ (g*h).
    Therefore you have a group monomorphism f: G → Sym(G) with f(g) := f_g, i.e. an embedding of G into Sym(G) that respects the group operation.
     
    Last edited: Nov 4, 2015
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