Finite projective planes admitting a projective linear group \(\operatorname{PSL}(2, q)\) (Q817641)

A linear space \(\mathcal S\) is a set \(\mathcal P\) of points, together with a set \(\mathcal L\) of distinguished subsets called lines such that any two points lie on exactly one line. The groups of automorphisms of linear spaces which are line-transitive (and so, point-transitive by Block's theorem [\textit{R. E. Block}, Math. Z. 96, 33--49 (1967; Zbl 0163.42304)]) have greatly been considered by a large number of people. The present paper continues in that tradition and the following result is proved. Theorem. Let \(G\leq\) Aut\((\mathcal S)\), where \(\mathcal S\) is a projective plane, and PSL\((2, q)\leq G\leq \text{P}\Gamma \text{L}(2, q)\) with \(q > 3\). If \(G\) acts point-transitively on \(\mathcal S\), then \(q = 7\) and \(\mathcal S\) is of order \(2\).