Let $G$ be a non-abelian simple group. I wonder if $G$ is complete; i.e., $\mathrm{Inn}\,G = \mathrm{Aut}\,G$.
Although I am an elementary learner, I know, just by simple calculation, that $\mathrm{Aut}\,G$ is complete: $\mathrm{Inn}\,\mathrm{Aut}\,G = \mathrm{Aut}\,\mathrm{Aut}\,G$. But, what can I do next?
I found some web sites saying $G$ is complete, so I think it may be true. But I have no idea to show the statement.
Could you give me a proof, or books in which I can find a proof?
It is not true; the outer automorphism of a finite simple group is not always trivial, for example the outer autmorphism of $A_n, n\neq 6$ is $\mathbb{Z}/2$ and $A_n$ is simple for $n>4$.
https://en.wikipedia.org/wiki/Outer_automorphism_group#In_finite_groups