Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras

Abstract

The question of existence of outer automorphisms of a simple algebraic group $G$ arises naturally both when working with the Galois cohomology of $G$ and as an example of the algebro-geometric problem of determining which connected components of $Aut(G)$ have rational points. The existence question remains open only for four types of groups, and we settle one of the remaining cases, type ^3$D$_4. The key to the proof is a Skolem-Noether theorem for cubic étale subalgebras of Albert algebras which is of independent interest. Necessary and sufficient conditions for a simply connected group of outer type $A$ to admit outer automorphisms of order 2 are also given.