Symmetric designs with translation blocks. I, II (Q788272)

A translation block of a symmetric 2-(v,k,\(\lambda)\) design is a block for which the group of automorphisms fixing every of its points acts transitively on the points not incident with it. The author proves that every non-trivial symmetric design with translation blocks has the same parameters as the system \({\mathcal P}_{n,q}\) of points and hyperplanes of PG(n,q). Moreover, if the design possesses a set of translation blocks intersecting in one point then it is isomorphic to a \({\mathcal P}_{n,q}\). The author then presents three contruction methods for symmetric designs with translation blocks by attention of other designs. In part II the author shows that every symmetric design with more than one translation block can be obtained from \({\mathcal P}_{n,q}\) by a combination of his construction methods. Thus he gets a generalisation of the Skornyakov-San Soucie theorem [\textit{D. R. Hughes} and \textit{F. C. Piper}, Projective planes (1973; Zbl 0267.50018)], which treats the case \(\lambda =1\), i.e. the case in which the design is a translation plane.