A characterization of the sharply 3-transitive finite permutation groups
Let G be a sharply 3-transitive permutation set on the finite set E. Let H be the subset of G consisting of all permutations having precisely two fixed points on E. The condition (*) is fulfilled for H if: \(h,k\in H\cup \{id_ E\}\Rightarrow hkh\in G.\) Making use of \textit{M. J. Kallaher}'s theorem on finite Bol quasifields [Algebras Groups Geom. 2, 300-312 (1985; Zbl 0589.51005)], the authors generalize a result by \textit{N. Percsy} [Geom. Dedicata 10, 269-282 (1981; Zbl 0454.51005)] by proving the following beautiful theorem: let \(| G| =q+1\), q odd, \(q\not\in \{9,5^ 2,7^ 2,11^ 2,19^ 2,29^ 2,59^ 2,3^ 4\}\), with the property that the identical permutation \(id_ E\) belongs to G and such that the condition (*) is fulfilled for the subset H of G. Then G is a permutation group and can thus be identified either with PGL(2,q) or with the group M(q) of all fractional semilinear transformations of the form \(z\mapsto (az+b)/(cz+d)\) if ad-bc is a square in \(GF(q)^*\) resp. of the form \(z\mapsto (a\sigma (z)+b)/(c\sigma (z)+d)\) if ad-bc is a non-square in \(GF(q)^*\), where \(\sigma\) denotes the unique automorphism of order 2 of GF(q). In a final remark, the authors communicate that in a separate note they are able to show that the restriction on q in the previous theorem can be reduced to \(q\neq 3^ 4\).
scientific article; zbMATH DE number 4156739
scientific article