TITLE: Distinguishing Inner Automorphisms
QUESTION [16 upvotes]: Given an automorphism $f$ of a group $G$, is there some way to determine whether or not $f$ is inner? What are some techniques for doing this?
If $G$ is finite, this can clearly be done by brute force. This isn't a very insightful answer though, and not the sort of answer I'm hoping to get. Rather, I would like to acquire a better understanding of what properties distinguish inner automorphisms from other automorphisms.

REPLY [4 votes]: There is an interesting theorem of Gary Seitz and Walter Feit (1984): Let $G$ be a finite simple group and let $\alpha \in \operatorname{Aut}(G)$. Suppose that $\alpha[C] = C$ for all conjugacy classes $C$ of $G$. Then $\alpha \in \operatorname{Inn}(G)$. The proof uses the classification of finite simple groups ...