Solvable linear groups on vector spaces whose dimension is the product of two primes

Abstract: A translation plane of dimension d over its kernel GF(q)= K can be represented in terms of a vector space of dimension 2d over K. The lines through the origin (zero vector) form a spread a class of mutually disjoint (except for the zero vector) d-dimensional subspaces which cover the vector space. The linear translation complement is the group of linear translations which acts as a collineation group of the plane. Not many different kinds of examples are known for the case where d and q are both odd. Except for Hering's plane of order 27, all of the examples known to the author have solvable linear translation complements and are either reducible or have a pair of lines through the origin in an orbit of length two. The first case includes the semi-field planes and some of the generalized Andr6 planes. The other generalized Andr6 planes of odd dimension fall in the second category. There must be some more examples, or restrictions on the nature of the linear translation complement, or both. The author has developed some restrictions when the group is non-solvable [7] or contains affine homologies [6]. In this paper, we examine the situation where the dimension over the kernel is an odd prime and we have an irreducible solvable subgroup of the linear translation complement. Many finite incidence structures are closely related to vector spaces over finite fields and the automorphisms (collineations) of these structures can be represented by linear groups i.e. groups of non-singular linear transformations. Frequently one knows that certain collineation groups must be subgroups of GL(n, q) for some specified n .and q. It would be helpful if one knew all of the subgroups of GL(n, q). This is essentially known for complex linear groups of