Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.