Combinatorial geometries representable over GF(3) and GF(q). II: Dowling geometries
[For part I see the first author, Discrete Comput. Geom. 5, No.1, 83-95 (1990; Zbl 0697.51007).] Let \(\{v_ 1,v_ 2,...,v_ n\}\) be a basis of an n-dimensional vector space of GF(3). The ternary Dowling geometry \(Q_ n(GF(3)^*)\) \((Q_ n\) for short) is the ternary geometry of rank n consisting of the points \(v_ 1,v_ 2,...,v_ n\) and the points \(v_ i-v_ j\) and \(v_ i+v_ j\), where \(i<j\). The Dowling geometry \(Q_ n\) contains \(n^ 2\) points and is representable over any field whose multiplicative group contains the cyclic group of order 2 as a subgroup. In the present paper the authors show that, with the exception of \(n=3\), a rank n geometry that is representable over GF(3) and GF(q) and contains exactly \(n^ 2\) points is isomorphic to the rank n Dowling geometry based on the multiplicative group of GF(3).
scientific article; zbMATH DE number 4151236