Chevalley groups of type $G_2$ as automorphism groups of loops

Details Chevalley groups of type $G_2$ as automorphism groups of loops Chevalley groups of type $G_2$ as automorphism groups of loops

2007-01-24

arxiv.org/abs/math/0701699v1

proceedings of Groups St Andrews 2001 in Oxford, Volume II, published in London Mathematical Society Lecture Note Series, 305,

Mathematics

Group Theory

13 pages

Scientific paper

Let $M^*(q)$ be the unique nonassociative finite simple Moufang loop constructed over $GF(q)$. We prove that $Aut(M^*(2))$ is the Chevalley group $G_2(2)$, by extending multiplicative automorphism of $M^*(2)$ into linear automorphisms of the unique split octonion algebra over GF(2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a split octonion algebra can be written as a sum of two elements of norm one.

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