Parabolic unitals in a family of commutative semifield planes》.pdfVIP

Discrete Mathematics 338 (2015) 1300–1306 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: /locate/disc Parabolic unitals in a family of commutative semifield planes Yue Zhou ∗ College of Science, National University of Defense Technology, Yanwachi Street No. 137, 410073 Changsha, China a r t i c l e i n f o a b s t r a c t Article history: In the commutative semifield planes constructed in Zhou and Pott (2013), we obtain a fam- Received 14 October 2014 ily of parabolic transitive unitals. For any unital in this family, we prove that there is a Received in revised form 22 February 2015 collineation group fixing it and acts sharply transitively on its affine points. We also con- Accepted 24 February 2015 sider its dual unital, the collinearity of its feet and show that as a design it is always resolv- Available online 16 March 2015 able. In particular, we give a necessary and sufficient condition under which a unital in our family is equivalent to a Ganley unital derived from a unitary polarity. Keywords: Unital © 2015 Elsevier B.V. All rights reserved. Semifield plane Collineation group 1. Introduction Let q be an integer and q ≥ 3. A unital of order q is a 2-(q3 + 1, q + 1, 1) design. For any unital U, if there exists a projective plane Π of order q2 such that U can be embedded in it, we call U an embedded unital. It is not difficult to see that each