Linear Geometries of Baer subspaces
نویسندگان
چکیده
In PG(2, q), q a prime power, we study the set T of Baer subplanes that contain a fixed triangle PQR. To construct a linear rank 2–geometry over T , we determine the dihedral groups, their orders and possible extensions that are generated by the involutions of two Baer subplanes of T . If q+1 is an odd prime, the (q+1)2 Baer subplanes through the triangle PQR are the points of an affine plane Aq+1 of order q + 1. Since it is necessary for this construction that q+1 is an odd prime, the change of the order from q to q+1 occurs only for q = 2 r , r ∈ IN, i.e. for the Fermat primes. Coordinatizing the affine plane Aq+1, we show that Aq+1 is desarguesian. Finally, we generalize the construction of AG(d, q + 1) out of PG(d, q2) to dimensions d ≥ 2 constructing the corresponding vector space.
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