Doubly transitive, bilinear dimensional dual hyperovals: The Conclusion

Abstract

In Yoshiara (2009) [17] determined possible groups which can act as a doubly transitive group of automorphisms on a dimensional dual hyperoval. The non-solvable, doubly transitive dual hyperoval were classified in Dempwolff (2018). By Yoshiara (2009) solvable, doubly transitive dual dimensional hyperovals are dual hyperovals defined over F2. If such a hyperoval has rank n, the doubly transitive group has the form G=E⋅H (semidirect product), E is a normal, elementary abelian group of order 2n in G and H≤ΓL(1,2n) acts transitively by conjugation on the non-trivial elements of E. The solvable, doubly transitive, dual hyperovals have been classified by Yoshiara (2008) and the author (Dempwolff 2015) provided the dual hyperoval is bilinear and GL(1,2n)≤H. In particular the doubly transitive, bilinear dual hyperovals are known if (n,2n−1)=1. If (n,2n−1)≠1 the group ΓL(1,2n) contains subgroups H, such that GL(1,2n)⁄≤H and H acts transitively by conjugation on the non-trivial elements of E. Our paper addresses this case and closes the investigation of doubly transitive, bilinear dual hyperovals.