Triple systems in \(\text{PG}(2,q)\) (Q1182931)

Let us call a set of \(n+1\) points of the projective space \(\text{PG}(n,q)\) a span. According to a result of \textit{S. A. Vanstone} and \textit{P. J. Schellenberg} [Vector spaces, geometries and designs, Util. Math. 6, 337-341 (1974; Zbl 0294.05014)] the points of \(\text{PG}(n,q)\) with spans as blocks form a block design which is denoted by \(\text{VS}(n,q)\). Therefore the triangles (spans) of \(\text{PG}(2,q)\) form a triple system \(\text{VS}(2,q)\) with parameters \(v=q^ 2+q+1\) and \(\lambda=q^ 2\). In this case the automorphism group of \(\text{VS}(2,q)\) is isomorphic to the group \(\text{P}\Gamma\text{L}(2,q)\) and contains a cyclic subgroup \(G\) of order \(q^ 2+q+1\) which is called the Singer group for \(\text{PG}(2,q)\). The authors prove that \(\text{VS}(2,q)\) contains a Steiner triple system \(\text{STS}(q^ 2+q+1)\) as a subdesign with \(G\) as a group of automorphisms. For the proof the authors represent \(\text{PG}(2,q)\) by the cyclic planar difference set in \(G\) and then show the existence of a set of triangles of \(\text{PG}(2,q)\) which is invariant under \(G\) and which forms a Steiner system on the points of \(\text{PG}(2,q)\). At the end of the paper the authors raise several interesting open question related to Steiner triple systems on \(q^ 2+q+1\) points.