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 the automorphism group of $G$ have rational points. The existence question remains open only for four types of groups, and we settle one of the remaining cases, type $^3D_4$. The key to the proof is a Skolem-Noether theorem for cubic etale 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.