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2018 | Buch

Latin Squares Based on Groups Kapitel lesen Erstes Kapitel lesen

Orthogonal Latin Squares Based on Groups

verfasst von: Anthony B. Evans

Verlag: Springer International Publishing

Buchreihe : Developments in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hall–Paige conjecture. The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry.
The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings. From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hall–Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems.
Expanding the author’s 1992 monograph, Orthomorphism Graphs of Groups, this book is an essential reference tool for mathematics researchers or graduate students tackling latin square problems in combinatorics. Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theory—more advanced theories are introduced in the text as needed.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter
Chapter 1. Latin Squares Based on Groups
Abstract
Latin squares and orthogonal Latin squares have been used in the construction of many classes of designs: nets, affine planes, projective planes, and transversal designs, in particular. When orthogonal sets of Latin squares are obtained from the multiplication table of a finite group by permuting columns, each square is determined by its first row, which is a permutation of the elements of the group. This enables us to describe and study orthogonality from a purely algebraic point of view, using difference matrices, complete mappings, and orthomorphisms. The nets, affine planes, projective planes, and transversal designs constructed in this way are characterized by the action of the group on these designs. We introduce these concepts in this chapter.
Anthony B. Evans
Chapter 2. When Is a Latin Square Based on a Group?
Abstract
A question that presents itself to us is, given a Latin square, is it isotopic to the Cayley table of a group? Several approaches have been proposed for answering this question, dating back to the 1800s. Some of the answers that we will present will be based on configurations in the given square or given bordered square, the quadrangle criterion for Cayley tables of quasigroups in general and its variants for loops, and the Thomsen condition characterizing Latin squares based on abelian groups. We will present the rectangle rule, a criterion for the normal multiplication table of a loop to be based on a group; and we will present approaches that are based on permuting rows and/or columns of the given square, or that treat rows and/or columns of the given square as permutations. Our emphasis will be on particular properties of Latin squares that characterize Latin squares based on groups. We will not emphasize algorithmic solutions, or algebraic properties of loops or quasigroups that characterize loops and quasigroups that are isotopic to groups.
Anthony B. Evans

Admissible Groups

Frontmatter
Chapter 3. The Existence Problem for Complete Mappings: The Hall-Paige Conjecture
Abstract
Given a group, does its Cayley table have an orthogonal mate? Equivalently, does a given group admit complete mappings? In this chapter we will introduce the existence problem for complete mappings of groups, a problem that we will solve for groups of odd order, countably infinite groups and finite abelian groups. For finite groups in general, we will present Hall and Paige’s important result: if the Sylow 2-subgroup of a finite group is nontrivial and cyclic, then the group does not admit complete mappings. Hall and Paige conjectured the converse. Work on this conjecture will be described in subsequent chapters, culminating in a proof of the Hall-Paige conjecture.
Anthony B. Evans
Chapter 4. Some Classes of Admissible Groups
Abstract
We have called a group admissible if it admits complete mappings and we proved that a finite group cannot be admissible if its Sylow 2-subgroup is nontrivial and cyclic. We proved the converse, known as the Hall-Paige conjecture, for abelian groups and groups of odd order. In this chapter we will prove the Hall-Paige conjecture true for more classes of groups. We will introduce HP-systems, an important tool in the construction of complete mappings, and use these to prove the Hall-Paige conjecture true for alternating and symmetric groups. We will also prove the Hall-Paige conjecture true for solvable groups, for Mathieu groups, for Suzuki groups, for certain unitary groups, and for groups whose Sylow 2-subgroups intersect trivially.
Anthony B. Evans
Chapter 5. The Groups GL(n, q), SL(n, q), PGL(n, q), and PSL(n, q)
Abstract
In this chapter we gather together admissibility proofs for the general linear group GL(n, q), the special linear group SL(n, q), the projective general linear group PGL(n, q), and the projective special linear group PSL(n, q). A group is admissible if it admits complete mappings. In this chapter we will show that the groups GL(n, q), SL(n, q), PGL(n, q), and PSL(n, q) are admissible when q is even, except when n and q are both 2. In this exceptional case, each of these groups is isomorphic to the nonabelian group of order 6, a group with a nontrivial, cyclic Sylow 2-subgroup, which therefore is not admissible by a result of Hall and Paige. In this chapter we will also prove the admissibility of GL(n, q), SL(2, q), PSL(2, q), and PGL(n, q) when q is odd.
Anthony B. Evans
Chapter 6. Minimal Counterexamples to the Hall-Paige Conjecture
Abstract
If the Hall-Paige conjecture is false, what might a minimal counterexample look like? This question was first raised by Aschbacher in 1990. He proved that any minimal counterexample must be “close” to being a simple group. In 1992 Evans tried to improve on this by extending complete mappings of H, a subgroup of G of index 2, to complete mappings of G; and complete mappings of G∕H, \(H\cong \mathbb {Z}_2\), to complete mappings of G. Evans’ proofs require that certain technical conditions hold for the extensions to work. In 2001 Dalla Volta and Gavioli improved on Aschbacher’s reduction and in 2009, using elementary techniques, Wilcox showed that any minimal counterexample to the Hall-Paige conjecture must be a nonabelian finite simple group. We will present these reductions in this chapter.
Anthony B. Evans
Chapter 7. A Proof of the Hall-Paige Conjecture
Abstract
In 2009 Wilcox proved that any minimal counterexample to the Hall-Paige conjecture must be a finite nonabelian simple group. He further proved that no finite simple group of Lie type, with the possible exception of2F4(2)′, the Tits group, could be a minimal counterexample to this conjecture. As the alternating groups were proved to be admissible in 1955 by Hall and Paige, and the Mathieu groups were proved admissible in 1993 by Dalla Volta and Gavioli, this left 22 possible minimal counterexamples to the Hall-Paige conjecture. Building on Wilcox’s work, Evans reduced the number of possible minimal counterexamples to the Hall-Paige conjecture to just one; Janko’s fourth group, J4. This last group was shown not to be a minimal counterexample by Bray, thus completing a proof of the conjecture. In this chapter we will cover proofs that finite nonabelian simple groups are admissible.
Anthony B. Evans

Orthomorphism Graphs of Groups

Frontmatter
Chapter 8. Orthomorphism Graphs of Groups
Abstract
In Chapter 1 we defined orthomorphisms and orthogonality of orthomorphisms. As orthogonality is a symmetric relation, we can define and study orthomorphism graphs: graphs whose vertices are orthomorphisms, and in which adjacency implies orthogonality. In this chapter we introduce orthomorphism graphs of groups. We describe the main problems of interest in the study of orthomorphism graphs, and we describe automorphisms and congruences of orthomorphism graphs. We describe some classes of orthomorphism graphs: the orthomorphism graph \(\mathcal {P}(G)\), composed of orthomorphisms of the form x↦xr, orthomorphism graphs derived from difference sets, orthomorphism graphs obtained from automorphisms, and orthomorphism graphs whose vertices are strong complete mappings.
Anthony B. Evans
Chapter 9. Elementary Abelian Groups. I
Abstract
Elementary abelian groups can be thought of as additive groups of finite fields. As such, all of the tools of field theory are available to us in the study of orthomorphism graphs of these groups. In particular, any function from a finite field to itself, and thus any orthomorphism of the additive group of the field, can be realized as a polynomial function. Several interesting classes of orthomorphisms will be described as sets of orthomorphism polynomials by placing restrictions on the polynomials. Classes of normalized orthomorphisms of additive groups of finite fields will be defined and studied using multiplication. The simplest such class is the class of linear orthomorphisms: these are of the form x↦ax. It is clear that such a mapping is an orthomorphism if and only if a≠0, 1. In this chapter we will also study quadratic orthomorphisms: these map x to ax if x is a square and to bx if x is a nonsquare.
Anthony B. Evans
Chapter 10. Elementary Abelian Groups. II
Abstract
We have used permutation polynomials in the study of orthomorphisms and orthomorphism graphs of elementary abelian groups. We have also studied linear orthomorphisms, orthomorphisms of the form x↦ax; and quadratic orthomorphisms, orthomorphisms of the form x maps to ax if x is a square and bx if x is a nonsquare. In this chapter we generalize linear and quadratic orthomorphisms by partitioning the elements of a finite field into cyclotomy classes {Ci} and then defining and studying classes of orthomorphisms consisting of mappings of the form x↦aix, if x ∈ Ci: this is the class of cyclotomic orthomorphisms. We will also show how classes of orthomorphisms of elementary abelian groups can be found by solving systems of linear equations called (σ, ε)-systems.
Anthony B. Evans
Chapter 11. Extensions of Orthomorphism Graphs
Abstract
In previous chapters we considered orthomorphisms and orthomorphism graphs of elementary abelian groups. Other than these groups, the other classes of abelian groups that have received significant attention are the cyclic groups and direct products of elementary abelian groups. In this chapter we will define the extension of the orthomorphism graph of a group G by a group H: this is an orthomorphism graph of G × H. We will discuss two special cases: H = GF(3)+ and G = GF(q)+ and \(H=\mathbb {Z}_3\) and \(G=\mathbb {Z}_m\), \(\gcd (m,3)=1\). We will use the 1978 construction of a set of four MOLS of order 15 by Schellenberg, Van Rees, and Vanstone to introduce extensions of orthomorphism graphs. We will discuss extensions by GF(3)+, and we will use difference equations to study the extension of Orth(GF(q)+) by GF(3)+. We will consider the special case of extensions of the orthomorphism graph of GF(q)+ consisting of linear orthomorphisms of GF(q)+. We will extend these methods to the study of extensions of cyclic groups by GF(3)+.
Anthony B. Evans
Chapter 12. ω(G) for Some Classes of Nonabelian Groups
Abstract
Almost all attempts to improve the lower bounds for ω(G) have been for abelian groups, in particular direct products of elementary abelian groups. The only classes of nonabelian groups for which attempts have been made to improve the lower bound for ω(G) are the dihedral groups and some of the linear groups of even characteristic. We will present these improvements in this chapter. We will derive improved lower bounds for ω(D2n), D2n the dihedral group of order 2n: these improvements will be obtained from improved lower bounds for \(\omega (\mathbb {Z}_n)\) and improved lower bounds for ω(D2n) when n is a power of 2. While proofs of admissibility show that ω(G) ≥ 1 when G is nonsolvable, very little work has been done on improving this bound. We will show that \(\omega (G)\ge \min \{\omega (\mathbb {Z}_{q-1}),\omega (\mathbb {Z}_{q+1})\}\), when G = GL(2, q) or SL(2, q), q even, q≠2. To establish this bound, we will use partitions of the element sets of the groups. For higher dimensions we will establish lower bounds for ω(GL(n, q)), q even, q≠2, using even-odd decompositions and partitions of the sets of odd-order elements of the groups.
Anthony B. Evans
Chapter 13. Groups of Small Order
Abstract
In this chapter we will describe what is known about the structure of Orth(G) for G a group of small order. Much of this knowledge is the result of computer searches. We will summarize known values and bounds for |Orth(G)| and ω(G) when |G|≤ 23. We will see that order 23 marks a divide. We can find values of |Orth(G)| in the literature for every group of order 23 or less, but we can find no such value for any group of order 24, other than the cyclic group of order 25 and those that have cyclic Sylow 2-subgroups and hence do not admit orthomorphisms. It will be seen that we know ω(G) for all abelian groups G of order 19 or less but can only establish a lower bound for \(\omega (\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_5)\); and we can only establish lower bounds for ω(G) for nonabelian groups G of order 16. We will also present what is known for groups of order greater than 23. Work on these larger groups has been concentrated on trying to improve lower bounds for ω(G) for groups G that are direct products of elementary abelian groups.
Anthony B. Evans

Additional Topics

Frontmatter
Chapter 14. Projective Planes from Complete Sets of Orthomorphisms
Abstract
From a complete set of orthomophisms of a group G of order n, we can construct a complete set of mutually orthogonal Latin squares of order n and a projective plane of order n. In this chapter we will describe the relationship between the complete sets of orthomorphisms and the corresponding projective planes. We will introduce Cartesian projective planes and show that these are the projective planes constructed from complete sets of orthomorphisms. We will describe the known classes of finite Cartesian projective planes. In addition, we will discuss nonexistence results for finite Cartesian projective planes. In particular we will prove the nonexistence of certain generalized Hadamard matrices, from which the nonexistence of complete sets of orthomorphisms of certain groups can be derived. We will also study Cartesian projective planes of prime order.
Anthony B. Evans
Chapter 15. Related Topics
Abstract
In this chapter we will introduce several topics, that have not been covered elsewhere in this book, but that are closely related to complete mappings and orthomorphisms of groups. In our discussion of these topics, we will outline the work that has been done, presenting many of the results without proofs. Our emphasis will be on the role played by orthomorphisms and related mappings. We will introduce classes of complete mappings and orthomorphisms as well as mappings that are closely related to complete mappings and orthomorphisms, in particular near complete mappings and near orthomorphisms. We will also discuss neofields, group sequencings and its variations, and starters.
Anthony B. Evans
Chapter 16. Problems
Abstract
In the 1992 book, “Orthomorphism graphs of groups”, we gave a list of 74 problems. Some of these have since been solved. In this chapter we will reorganize, update, and amend this list of problems and will include many additional problems.
Anthony B. Evans
Backmatter
Metadaten
Titel
Orthogonal Latin Squares Based on Groups
verfasst von
Anthony B. Evans
Copyright-Jahr
2018
Electronic ISBN
978-3-319-94430-2
Print ISBN
978-3-319-94429-6
DOI
https://doi.org/10.1007/978-3-319-94430-2

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