A generalized homology of a generalized hexagon 5^ is an auto-morphism of S? fixing all points on two mutually opposite lines or fixing all lines through two mutually opposite points. We show that if 5? is finite and if it admits "many" generalized homologies, then 5? is Moufang and hence classical. 1. Introduction and notation, A (finite thick) generalized hexagon of order (s, t) is a point-li...

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 s...

In this paper, we classify all generalized quadrangles weakly embedded of degree 2 in projective space. More exactly, given a (possibly infinite) generalized quadrangle Γ = (P,L, I ) and a map π from P (respectively L) to the set of points (respectively lines) of a projective space PG(V ), V a vector space over some skew field (not necessarily finite-dimensional), such that: (i) π is injective ...

Let # be the exceptional 27-dimensional Jordan algebra over C. Its automorphism group is the Lie group F4(@) and this group is known to have a finite subgroup AL, where A is a self centralizing elementary abelian of order 27, L g SL(3, 3), and L normalizes A. As an A-module, 2 decomposes into a direct sum of l-dimensional spaces jrwhich afford the 27 distinct linear characters x E A n := Hom(A,...