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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