In this paper, we consider finite flag-transitive affine planes with asolvable automorphism group. Under a mild number-theoretic condition involvingthe order and dimension of the plane, the translation complement must contain alinear cyclic subgroup that either is transitive or has two equal-sized orbitson the line at infinity. We develop a new approach to the study of such planesby associating them with planar functions and permutation polynomials in theodd order and even order case respectively. In the odd order case, wecharacterize the Kantor-Suetake family by using Menichetti's classification ofgeneralized twisted fields and Blokhuis, Lavrauw and Ball's classifcation ofrank two commutative semifields. In the even order case, we develop a techniqueto study permutation polynomials of DO type by quadratic forms and characterizesuch planes that have dimensions up to four over their kernels.
展开▼
