Flag-transitive and almost simple orbits in finite projective planes (Q993641)

Let \(\Pi\) be a projective plane of order \(n\) admitting a collineation group \(G\) which acts doubly transitively (flag transitively) on a point-subset \(\mathcal O\) of size \(v\). Then \(\mathcal O\) is either (i) a non-trivial 2-\((v, k, 1)\) design, (ii) an arc, or (iii) a subset of a line. The paper focuses on the case (i) under the assumptions that \(G\) induces a flag-transitive and almost simple automorphism group on \(\mathcal O\) and \(n \leq \sum ({\mathcal O}) = b + v + r + k\), where \(b\) denotes the number of blocks of \(\mathcal O\) and \(r\) the number of blocks of \(\mathcal O\) incident with any given point of \(\mathcal O\). The author arrives (Theorem 1.1) at ten distinct cases when \(G\) acts faithfully on \(\mathcal O\) and two cases when this action is non-faithful, either ``regime'' being characterized by one open case. Reviewer's remark: In case (2a) of Theorem 1.1 there is an obvious misprint: ``75'' should read ``57''.