Unitals Admitting All Translations

Abstract

Among all 2‐(q3+1,q+1,1)‐designs, we characterize the Hermitian unitals by the existence of sufficiently many translations. In arbitrary 2‐(q3+1,q+1,1)‐designs, each group of translations with given center acts semiregularly on the set of points different from the center.

Finite elation Laguerre planes admitting a two-transitive group on their set of generators

“…The remaining cases (PSL(2, q), unitary groups, Ree groups, and the socle PSL (2,8) of the smallest Ree group R(3)) are treated in a separate paper [31]. There we show that only the groups PSL(2, q) actually occur, and the corresponding Laguerre planes are Miquelian.…”

“…The group PΓL (2,8) induced by the stabilizer of a circle in the Miquelian plane of order 8 has elementary abelian Sylow 2-subgroups, and contains no elements of order 4. Thus the present case is indeed impossible.…”

“…All of this indicates that elation Laguerre planes form a nice subclass of Laguerre planes and that, if there are finite non-ovoidal Laguerre planes, elation Laguerre planes certainly are a natural class to look for them. Doubly transitive groups of automorphisms have been investigated for various geometries, see for example [6], [23], [11], [9], [13], [8], [36, 8.5]. In this note we follow the program for finite translation planes to construct such planes from information about a suitable group in the translation complement of the collineation group and to classify all arising planes.…”

A non-classical unital of order four with many translations

“…As the classical unitals are characterized by the fact that they have non-collinear translation centers (see [9]), the full group of automorphisms of U E coincides with the block stabilizer L.…”

“…The main result of [9] states that the unital U is classical (i.e., isomorphic to the hermitian unital corresponding to the field extension F q 2 /F q ) if it has non-collinear translation centers. Unitals with precisely one translation center seem to exist in abundance (we indicate several quite different classes of examples in Section 5 below).…”