Automorphisms of Pairwise Combinatorial Designs

Abstract

In this thesis, we investigate group actions on certain families of pairwise combinatorial designs, in particular Hadamard matrices and symmetric 2-(4t-1, 2t-1, t-1) designs. A Hadamard matrix H is called cocyclic if a certain quotient of the automorphism group contains a subgroup acting regularly on the rows and columns of H. Cocyclic Hadamard matrices (CHMs) were first investigated by de Launey and Horadam in the early 1990s. We develop an algorithm for constructing all CHMs of order 4t based on a known relation between CHMs and relative difference sets. This method is then used to produce a classification of all CHMs of order less than 40. This is an extension and completion of work of de Launey and Ito. Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon. We then use our classification to show that the only cocyclic Hadamard matrices with non-affine automorphism group are those that arise from the Paley Hadamard matrices. As a corollary of this result, we show that twin prime power difference sets and Hall sextic residue difference sets each give rise to a unique CHM. If H is a CHM developed from a difference set then the automorphism group of H is doubly transitive. We classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group. A key component of this is a complete list of difference sets corresponding to the Paley Hadamard matrices. As part of our classification we uncover a new triply infinite family of skew- Hadamard difference sets. To our knowledge, these are the first skew-Hadamard difference sets to be discovered in non-abelian p-groups with no exponent restriction.