Automorphism groups of simple Moufang loops over perfect fields

Abstract

Let $F$ be a perfect field and $M(F)$ the non-associative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then ${\rm Aut}(M(F))$ is equal to $G_2(F) \rtimes {\rm Aut}(F)$. In particular, every automorphism of $M(F)$ is induced by a semilinear automorphism of $O(F)$. The proof combines results and methods from geometrical loop theory, groups of Lie type and composition algebras; the result being an identification of the automorphism group of a Moufang loop with a subgroup of the automorphism group of the associated group with triality.