Eine neue Serie von endlichen Translationsebenen. (A new series of finite translation planes) (Q1110094)

For any odd prime power q the authors construct a translation plane of order q admitting non-trivial shears. For \(q=3,5,7\) there arise a desarguesian plane, the Betten-Walker-plane of order 25, and the plane of order 49 from the first author [Geom. Dedicata 27, No.1, 87-100 (1988)]. The construction uses a congruence which is induced from a flock of ruled quadrics in a 4-dimensional vector space V [cf. the first author, Geom. Dedicata 22, 235-245 (1987; Zbl 0609.51001)]. This flock is parametrized by a skilfully chosen set \({\mathcal L}\) of cardinality q. The authors study the collineation group \({\mathcal K}\) in detail. Via the symmetry group of \({\mathcal L}\) they determine for instance the point orbits of \({\mathcal K}_ 0\) \((:=stabilizer\) of \(0\in V)\), and its group-theoretical structure.