LOW DIMENSIONAL MODELS OF THE FINITE SPLIT CAYLEY HEXAGON 9 Then the two-point stabiliser of X and Y inside SU3 consists of elements MA with A of the form A = a b 0 d e 0 0 0 1 where (a + dω)ω = b + and ( a b d e )( aq dq bq eq ) = I. Let’s consider one of these elements MA. Then (Aq)T UA = 0 0 −aqω+dq 0 0 −bqω+eq −aωq+d −bωq+e 0 and we see that this matrix is a scalar multiple of U (the scalar being (−bωq + e)). Therefore MA fixes the Baer subgenerator defined by [U, ]. Hence there is a unique element of Ω on X and Y . 2.3. Classifying the suitable sets of Baer subgenerators. Theorem 2.6. Suppose Ω is a set of Baer subgenerators of H(3,q2) with a point in O, such that every affine point is on q +1 elements of Ω spanning a Baer subplane. Then Ω is an orbit under SU3. Proof. Let b be a Baer subgenerator of H(3,q2) with a point in O. If b is another Baer subgenerator of H(3,q2) with a point in O such that b and b meet in an affine point and span a fully contained Baer subplane, then we will show that there is some element of SU3 which maps b to b . Without loss of generality, we can choose our favourite Baer subgenerator and our favourite affine point. Suppose we have a fixed Baer subgenerator b giving the dual Baer subline defined by U = 0 0 −ω 0 0 1 −ωq 1 0 and on the generator = (1,ω, 0, 0), (0, 0, 1,ω) where N(ω) = −1. Let P be the affine point (0, 0, 1,ω) and consider an arbitrary generator on P where := (0, 0, 1,ω), (1,ν, 0, 0) and N(ν) = −1. Suppose we have a Baer subgenerator b on P , on the generator , defined by the matrix U . Since every element of P u O is in the dual Baer subline defined by U , we have that U can be written as a 0 β 0 a γ βq γq c where a GF(q) and β, γ GF(q2). For (1,ν, 0, 0) to be in the nullspace of U , we must have a = 0 and β = −γν. That is, U is just 0 0 −γν 0 0 γ −γqνq γq c . Now b and b span a fully contained Baer subplane if and only if the dual Baer sublines defined by U and U share only the points of P u O, on O. Indeed suppose, by way of contradiction, that there is a point Z of O in common between the dual Baer sublines defined by U and U . Then Zu meets b in a point Q, different from L and P and it meets b in a point Q different from L (L = π ∩b ) and P . Thus Zu ∩P u meets H(3,q2) in a Baer
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