Translation planes in which the stabilizer of every line is doubly transitive

Abstract

Ltineburg [12] recently investigated affine planes having a collineation group G with the property that the stabilizer of every affine line I is doubly transitive on L He shows that all such planes are translation planes and that G contains the group of translations. (This was proven independently in [10].) Such planes are generalizations of rank 3 affine planes which have been investigated by several authors. (See [11], p. 121.) In this article we will investigate the question of which translation planes ~ satisfy Liineburg's hypothesis. We restrict our attention to planes having odd dimension d over their kernel GF(q), q=p" with p an odd prime. Our results are generalizations of the known results for rank 3 affine planes. The ideas of the proofs are as follows: For an affine point 0 the group 6= Go has the property that for every component l ofzr (i.e., line through ~0) Gz is transitive on the points of l different than 0. Thus G~ induces a transitive group of semi-linear transformations on l considered as a vector space over the kernel of the plane. Applying results of Hering [5, 6] we can show that a Sylow u-subgroup of Gt, u a prime p-primitive divisor of qd_ 1, is normal in Gt. (It is here that we need the assumption concerning d and q. Most of our results, however, are still true if p=2 and we point out when this is so.) The main results are: (1) If G induces a regular permutation group on one of its orbits on the line Iv, then the only possible translation planes are semifield planes, generalized Andr6 planes, and Desarguesian planes. Moreover, in the case of the semi-field planes they must be rank 3 planes. (Theorems 2, 3, 4 and Corollary 4.1.) (2) If p> 5 and r is odd, then either ~ is a semifield plane, a generalized Andr6 plane, or r~ has an invariant Desarguesian decomposition. For many values ofp and r in the last case, ~ is either Desarguesian or a rank 3 plane. (Corollary 5.1 and the Remark following.)