3 research outputs found
Embeddings of Ree unitals in a projective plane over a field
We show that the Ree unital has an embedding in a projective
plane over a field if and only if and is a subfield of
. In this case, the embedding is unique up to projective linear
transformations. Besides elementary calculations, our proof uses the
classification of the maximal subgroups of the simple Ree groups
SL(2,q)-Unitals
Unitals of order are incidence structures consisting of points such that each block is incident with points and such that there are unique joining blocks. In the language of designs, a unital of order is a - design. An affine unital is obtained from a unital by removing one block and all the points on it. A unital can be obtained from an affine unital via a parallelism on the short blocks. We study so-called (affine) -unitals, a special construction of (affine) unitals of order where is a prime power. We show several results on automorphism groups and translations of those unitals, including a proof that one block is fixed by the full automorphism group under certain conditions. We introduce a new class of parallelisms, occurring in every affine -unital of odd order. Finally, we present the results of a computer search, including three new affine -unitals and twelve new -unitals
ON THE REE UNITAL
The embedding of a Ree Unital R(q) in a finite projective plane P of order up to q^4 is investigated when the Ree group is induced on R(q) by a collineation group of P. In particular, it is shown that such a embedding is not admissible for q>3, extending in this way a result of Lüneburg dating back to 1965