Extending automorphism groups to inner automorphism groups.

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In summary, the conversation discussed the concept of inner automorphisms and their relationship to normal subgroups. It was clarified that \phi_g, defined as \phi_g(h) = g h g^{-1}, is indeed an automorphism of N but may not necessarily be an inner automorphism if g is not guaranteed to be in N. The conversation also explored the possibility of extending a group to a larger group where all automorphisms are given by inner automorphisms from the larger group. It was concluded that this is possible through the use of a semidirect product, where the automorphisms of the original group are inner automorphisms of the larger group.
  • #1
Kreizhn
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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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  • #2
Hi Kreizhn! :smile:

Kreizhn said:
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...
 
  • #3
I shall have to go back and look at my notes on semi-direct products a bit more to understand this fully, but thanks.
 

1. What is the concept of automorphism groups?

Automorphism groups are mathematical structures that represent symmetries or transformations of an object onto itself. They are often used in algebraic structures such as groups, rings, and fields.

2. How do automorphism groups relate to inner automorphism groups?

Automorphism groups can be extended to inner automorphism groups by considering only the automorphisms that fix a specific element in the group. This results in a subgroup of the original automorphism group, known as the inner automorphism group.

3. Why is it important to extend automorphism groups to inner automorphism groups?

Extending automorphism groups to inner automorphism groups allows for a more detailed understanding of the symmetries and transformations within a mathematical structure. It also provides a way to study the structure of a group in a more specific and controlled manner.

4. How are inner automorphism groups used in practical applications?

Inner automorphism groups have various applications in fields such as cryptography, physics, and computer science. They are used in encryption algorithms, symmetry analysis of physical systems, and the study of automata and formal languages.

5. Are there any limitations to extending automorphism groups to inner automorphism groups?

One limitation is that not all automorphisms can be extended to inner automorphisms. This is because some automorphisms do not fix any elements in the group. Additionally, the process of extending automorphism groups to inner automorphism groups can be complex and time-consuming for larger groups.

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