Extending automorphism groups to inner automorphism groups.

[Kreizhn]

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

 

[micromass]

Hi Kreizhn!

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

That is correct.

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 G \to H such that \text{Aut}(G) \cong \text{Inn}(H) ?

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

\phi:T\rightarrow Aut(G)

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

\phi_h(g)=hgh^{-1}

So the automorphisms of G are inner automorphisms of the semidirect product...

 

[Kreizhn]

I shall have to go back and look at my notes on semi-direct products a bit more to understand this fully, but thanks.