2 4 Ja n 20 07 Chevalley Groups of Type G 2 as Automorphism Groups of Loops

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. Let C be a finite-dimensional vector space over a field k, equipped with a quadratic form N : C −→ k and a multiplicative operation ·. Following [7], we say that C = (C, N, +, ·) is a composition algebra if (C, +, ·) is a nonassociative ring with identity element e, N is nondegenerate, and N (u · v) = N (u)N (v) is satisfied for every u, v ∈ C. The bilinear form associated with N will also be denoted by N .