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I know that an automorphism of a group is just an isomorphism of that group with itself. I know that an inner automorphism is an isomorphism/automorphism of the form $\phi_g$ where $\phi_g(x) = gxg^{-1}$.

I have heard people talking about an outer automorphism. What are these?

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  • $\begingroup$ As far as I know, "outer" means "not inner" in this context. $\endgroup$ – saulspatz Mar 5 at 13:53
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    $\begingroup$ The usage is slightly confusing An outer automorphism is an element of $ {\rm Aut}(G) \setminus {\rm Inn}(G)$. But the outer automorphism group is defined to be the quotient group ${\rm Aut}(G)/{\rm Inn}(G)$. $\endgroup$ – Derek Holt Mar 5 at 13:54
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    $\begingroup$ Is it really necessary to ask when answers are so easy to find? An automorphism of a group which is not inner is called an outer automorphism. It also goes on to explain the issue Derek mentions. I don't want to sound antagonistic, but I do want to encourage you to do at least minimal research into your question before asking. We do not need to re-record everything that's easy to find on the internet again on our site... $\endgroup$ – rschwieb Mar 5 at 14:00
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    $\begingroup$ Just because an answer is easy to find, it does not follow that the answer is sensible. See my comment to the answer of @HagenvonEitzen. $\endgroup$ – Lee Mosher Mar 5 at 15:20
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    $\begingroup$ Possible duplicate of What is the outer automorphism group? $\endgroup$ – verret Mar 5 at 19:17
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An outer automorphism is just an automorphism that is not an inner automorphism. But be careful: We know that the group $\operatorname{Inn}(G)$ of inner automorphisms is a normal subgroup of the group $\operatorname{Aut}(G)$ of all automorphisms of $G$, and the quotient $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$ is called the outer automorphism group. So confusingly, the elements of the outer automorphism group are not outer automorhims, but rather equivalence classes of outer automorphisms ...

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    $\begingroup$ To be pedantic, the non-identity elements of the outer automorphism group are not outer automorphisms but rather ... $\endgroup$ – Derek Holt Mar 5 at 14:25
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    $\begingroup$ Although it appears that I am fighting a strong tide, I have to register a strong disagreement with this usage of the term "outer automorphism". Using it to mean "an automorphism that is not inner" is really a terrible usage. I would encourage anyone writing about outer automorphism groups to avoid that usage. $\endgroup$ – Lee Mosher Mar 5 at 15:17
  • $\begingroup$ This was really helpful. $\endgroup$ – John Doe Mar 5 at 15:18
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    $\begingroup$ It turns out that many people unfortunately say "outer automorphism" to mean "non-inner automorphism", and there is not even the excuse that "non-inner" is hard to write or pronounce. This is confusing, sure, but is hopelessly frequent. Just search "is an outer automorphism". $\endgroup$ – YCor Mar 5 at 15:44
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    $\begingroup$ @LeeMosher If we were to start over, wouldn't it make more sense to keep that usage, but rename $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ to something else. After all, why shouldn't an outer automorphism actually be an automorphism, rather than a coset? (I consider both usage more or less entrenched, so on equal footing if we're discussing alternate definitions.) $\endgroup$ – verret Mar 5 at 19:21

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