Groups and Inner Automorphisms

  1. 1. The problem statement, all variables and given/known data
    Let G be a group. Show that G/Z(G) [itex]\cong[/itex] Inn(G)


    3. The attempt at a solution
    G/Z(G) = gnZ(G) for some g ε G and for any n ε N
    choose some g-1 such that
    g(g-1h) = g(hg-1)
    and the same can be done switching the g and g-1

    This doesn't feel right at all...
     
  2. jcsd
  3. micromass

    micromass 19,188
    Staff Emeritus
    Science Advisor
    Education Advisor

    I don't see what this has to do with the problem??

    Can you find a surjective homomorphism

    [tex]f:G\rightarrow Inn(G)[/tex]

    and then apply the first isomorphism theorem?
     
  4. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    No, not right. Given an element g of G can you name an inner automorphism of G corresponding to g? When do two different elements of G, g1 and g2 give define the same automorphism?
     
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