Automorphisms of Pairwise Combinatorial Designs
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Date
2012-02-14Author
Ó Catháin, Padraig
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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.