Affine Groups with Two Self-Paired Orbitals

Abstract

In this paper we describe all known instances of primitive soluble groups of affine type which have a single self-paired nondiagonal orbital. We show how to construct representations of the point stabiliser that are imprimitive, semilinear, subfield, or tensor decomposable. As a corollary we give a constructive proof that there exist groups with arbitrarily large numbers of orbitals, only two of which are self-paired.

Key Words:

• Permutation group
• Primitive
• Affine
• Orbital

Acknowledgment

This work was written under the supervision of Professor P. J. Cameron. The author acknowledges the support of both the Carnegie Trust for the Universities of Scotland and the Australian Research Council.