Flag-transitive and almost simple orbits in finite projective planes

FINITE s-ARC TRANSITIVE CAYLEY GRAPHS AND FLAG-TRANSITIVE PROJECTIVE PLANES

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

Transitive hyperovals in finite projective planes

Transitive projective planes

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components. 1 Background definitions and main results We say that a projective plane is transitive (respectively primitive) if it ...

Nearly flag-transitive affine planes

Spreads of orthogonal vector spaces are used to construct many translation planes of even order q, for odd m > 1, having a collineation with a (q − 1)-cycle on the line at infinity and on each of two affine lines.

Finite flag-transitive affine planes with a solvable automorphism group

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...