Extending automorphism groups to inner automorphism groups.by Kreizhn Tags: automorphism, extending, groups 

#1
Jun2411, 11:41 AM

P: 743

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
Jun2411, 01:19 PM

Mentor
P: 16,542

Hi Kreizhn!
[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... 



#3
Jun2411, 01:30 PM

P: 743

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



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