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Using Isomorphism Theorem to show

  1. Apr 8, 2010 #1
    1. The problem statement, all variables and given/known data

    If G is a group and aϵG, then the inner automorphism θa: G --> G is defined by θa(g) = aga-1. Let Inn(G) = group of inner automorphisms and Z(G) = the centre of G.
    Use the Isomorphism theorem to show G/Z(G)Inn(G).

    2. Relevant equations



    3. The attempt at a solution
    Firstly, the inner automorphism θa: G --> G defined by θa(g) = aga-1 is a group homomorphism, with its kernel being the centre of G, denoted Z(G). And its image = Inn(G).
    So since the theorem says Im(θ) G/ker(θ), then G/Z(G)Inn(G).

    Would I have to somehow prove that Z(G) = ker(θ)? I don't really know where to begin.
    Likewise with Im(θ) = Inn(G).

    Please let me know if this attempt is sort of close to the actual solution.
    Thankss!!!
     
  2. jcsd
  3. Apr 9, 2010 #2
    You want to show that the assigment [tex] a\mapsto \theta_a [/tex] is a homomorphism from G to Aut(G) (the group of automorphisms of G). Clearly the image of this map is Inn(G). From what you wrote I think you were looking at an individual [tex]\theta_a[/tex] (which, btw, the kernel can be bigger than the center since there can be elements that commute with [tex]a[/tex] but not with everything, so the center will be a subgroup of the kernel of an individual [tex]\theta_a[/tex])
     
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