Extending automorphism groups to inner automorphism groups.

  1. I wanted to make clear just a quick technical thing. If G is a group, N is a normal subgroup, and [itex] \phi_g \in \text{Inn}(G), \phi_g(h) = g h g^{-1} [/itex] then [itex] \phi_g [/itex] is an automorphism of N, right? However, is it the case that we cannot say that [itex] \phi_g [/itex] is an inner automorphism, since we are not guaranteed that [itex] g \in N [/itex]? I think this is the case, but I just want to be clear.

    Furthermore, is it then possible to extend any group to a larger group such that all automorphisms are given by inner automorphisms from the larger group? More precisely, if G is a group, is there always an injective homomorphism [itex] G \to H [/itex] such that [itex] \text{Aut}(G) \cong \text{Inn}(H) [/itex] ?
     
  2. jcsd
  3. micromass

    micromass 18,537
    Staff Emeritus
    Science Advisor

    Hi Kreizhn! :smile:

    That is correct.

    Indeed, let G be our group we wish to extend. Let T be a group such that there exists an epimorphism

    [tex]\phi:T\rightarrow Aut(G)[/tex]

    (for example, take [itex]T=Aut(G)[/itex]), then the semidirect product [itex]G\rtimes_{\varphi} T[/itex] is an extension you're looking for. Indeed, automorphism of G has the form [itex]\phi(h):=\phi_h[/itex]. And by construction of the semidirect product, we have that for each g in G

    [tex]\phi_h(g)=hgh^{-1}[/tex]

    So the automorphisms of G are inner automorphisms of the semidirect product...
     
  4. I shall have to go back and look at my notes on semi-direct products a bit more to understand this fully, but thanks.
     
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