TITLE: What does an outer automorphism look like?
QUESTION [18 upvotes]: I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about finding this guy, but I have a larger conceptual question - what does an outer automorphism really look like? Is there an intuitive way to understand the difference between an inner and an outer automorphism? Inner automorphisms have always seemed easier for me to understand since we have an explicit representation $ghg^{-1}$ for members of the group. I can also understand this representation in terms of the Rubik's cube - rotate an edge, rotate a perpendicular edge, and the rotate the other edge back (is not the same as just rotating the perpendicular edge). What does an "outer automorphism" look like?

REPLY [3 votes]: Another rather accessible example of an outer automorphism can be found on matrix groups: Look at the inverse transpose map
$$\operatorname{GL}(n,F) \to \operatorname {GL}(n,F), \quad A \mapsto (A^{-1})^\top.$$
It is an automorphism, and except from all cases with $n=1$ and $\operatorname{GL}(2,\mathbb F_2)$, it is is outer.