Dane Flannery

Algebraic Design Theory

Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.

Cocyclic Hadamard matrices and difference sets

Abstract This paper locates cocyclic Hadamard matrices within the mainstream of combinatorial design theory. We prove that the existence of a cocyclic Hadamard matrix of order 4t is equivalent to the existence of a normal relative difference set with parameters (4t,2,4t,2t) . In the basic case we note there is a corresponding equivalence between coboundary Hadamard matrices and Menon–Hadamard difference sets. These equivalences unify and explain results in the theories of Hadamard groups, divisible designs with regular automorphism groups, and periodic autocorrelation functions.

Calculation of cocyclic matrices

Abstract In this paper we provide a method of explicitly determining, for a given finite group G and finitely generated G -module U trivial under the action of G , a representative for each element (2-cocycle class) in H 2 ( G , U ). These cocycles are naturally displayed as | G | × | G >; z . sfnc ; matrices. An example of calculating cocyclic matrices using the method is given.

Algorithms for computing with nilpotent matrix groups over infinite domains

We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical nilpotency testing algorithm for matrix groups over an infinite field. We also provide algorithms to answer a number of structural questions for a nilpotent matrix group.The main algorithms have been implemented in GAP, for groups over the rational number field.

Computing in Nilpotent Matrix Groups

We present algorithms for testing nilpotency of matrix groups over finite fields, and for deciding irreducibility and primitivity of nilpotent matrix groups. The algorithms also construct modules and imprimitivity systems for nilpotent groups. In order to justify our algorithms, we prove several structural results for nilpotent linear groups, and computational and theoretical results for abstract nil-potent groups, which are of independent interest.

On deciding finiteness of matrix groups

We provide a new, practical algorithm for deciding finiteness of matrix groups over function fields of zero characteristic. The algorithm has been implemented in GAP. Experimental results and extensions of the algorithm to any field of zero characteristic are discussed.

LINEAR GROUPS OF SMALL DEGREE OVER FINITE FIELDS

For n = 2,3 and finite field 𝔼 of characteristic greater than n, we provide a complete and irredundant list of soluble irreducible subgroups of GL(n,𝔼). The insoluble irreducible subgroups of GL(2,𝔼) are similarly determined. Each group is given explicitly by a generating set of matrices. The lists are available electronically.

Zariski density and computing in arithmetic groups

For

NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS

n > 2

CLASSIFICATION OF NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS

, let