Harriet Pollatsek

Difference sets : connecting algebra, combinatorics and geometry

Table of Contents:* Introduction * Designs * Automorphisms of designs * Introducing difference sets * Bruck-Ryser-Chowla theorem * Multipliers * Necessary group conditions * Difference sets from geometry * Families from Hadamard matrices * Representation theory * Group characters * Using algebraic number theory * Applications * Background * Notation * Hints and solutions to selected exercises * Bibliography * Index * Index of parameters

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D20 ×  Z2(for example, if G ≃ D20 × K). (2) G has a normal 5-Sylow subgroup and an elementary abelian 2-Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2-Sylow subgroup S and 5-Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D10image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillons “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work.

Opening Lines: An Introduction to the Volume

In this opening chapter, the editors set the stage for the wide-ranging description and discussion of work in mathematics education awaiting readers of this volume. They define how the phrase “work in mathematics education” is to be understood for this volume and explain how the 25 chapters are grouped according to intended beneficiaries of the work. The editors describe the genesis of the book: how the idea arose in June 2015 and how it was intended to be an extension of the conversation that would take place at the 2016 Joint Mathematics Meetings panel on “Work in Mathematics Education in Departments of Mathematical Sciences,” co-sponsored by the Association for Women in Mathematics (AWM) Education Committee and the American Mathematical Society Committee on Education. To entice the reader to explore the volume, the editors highlight some of the contents and note common themes and connections among the chapters. This chapter also summarizes the multi-stage process that brought the idea for this book to fruition so that the reader may understand the selection and peer review process. As many of the chapters do, this one closes with a final reflection by its authors on their involvement in this project.

Looking for Difference Sets in Groups with Dihedral Images

AbstractWe prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p

Permutationally invariant codes for quantum error correction

\tilde q

Automorphisms of designs

with p an odd prime, contains a nontrivial (v, k, λ) difference set D with order n = k − λ prime to p and self-conjugate modulo p. If G has an image of order p, then 0 ≤ 2a + ∈

Quantum Error Correction: Classic Group Theory Meets a Quantum Challenge

\sqrt n