Outer Automorphisms of Algebraic Groups and a Skolem-Noether Theorem for Albert Algebras

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.

2010 Mathematics Subject Classification: Primary 20G41; Secondary 11E72, 17C40, 20G15

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