Polarity and transitive parabolic unitals in translation planes of odd order
A unital \(U\) of size \(q^3 + 1\) embedded in a projective plane \(\pi\) of order \(q^2\) is called a transitive parabolic unital if the collineation group of \(\pi\) stabilizing \(U\) fixes one point of \(U\) and acts transitively on the remaining points of \(U\). In the paper under review the authors show that the commutative Dickson semifield plane \(\pi\) of odd order \(q^2\) contains a transitive parabolic unital \(U\) whose stabilizer in Aut(\(\pi)\) contains an affine homology and a non-commutative normal subgroup \(K\) which is sharply transitive on the affine points of \(U\). Coordinate arguments are used to prove this result. It has been conjectured that if a transitive parabolic unital \(U\) consists of the absolute points of a unitary polarity in a commutative semifield plane \(\pi\), as in the above case, then the sharply transitive normal subgroup \(K\) of the stabilizer of \(U\) in Aut\((\pi)\) must necessarily be non-commutative.
scientific article; zbMATH DE number 1890526