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Extending automorphism groups to inner automorphism groups.

by Kreizhn
Tags: automorphism, extending, groups
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Kreizhn
#1
Jun24-11, 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] ?
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micromass
#2
Jun24-11, 01:19 PM
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Hi Kreizhn!

Quote Quote by Kreizhn View Post
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.
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 [itex] G \to H [/itex] such that [itex] \text{Aut}(G) \cong \text{Inn}(H) [/itex] ?
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...
Kreizhn
#3
Jun24-11, 01:30 PM
P: 743
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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