# What is an outer automorphism? [closed]

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?

## closed as off-topic by Servaes, GNUSupporter 8964民主女神 地下教會, Dietrich Burde, verret, LeucippusMar 6 at 1:03

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• As far as I know, "outer" means "not inner" in this context. – saulspatz Mar 5 at 13:53
• 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 ${\rm Aut}(G)/{\rm Inn}(G)$. – Derek Holt Mar 5 at 13:54
• 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... – rschwieb Mar 5 at 14:00
• 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. – Lee Mosher Mar 5 at 15:20
• Possible duplicate of What is the outer automorphism group? – verret Mar 5 at 19:17

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 ...
• @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.) – verret Mar 5 at 19:21