Details

Orthogonal Latin Squares Based on Groups

Developments in Mathematics, Band 57

von: Anthony B. Evans
149,79 €
Verlag:	Springer
Format:	PDF
Veröffentl.:	17.08.2018
ISBN/EAN:	9783319944302
Sprache:	englisch

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Beschreibungen

Titelbeschreibung

<div>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.  </div><div><br></div><div>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.  </div><div><br></div><div>Expanding the author’s 1992 monograph, <i>Orthomorphism Graphs of Groups</i>, 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. </div><div><br></div>

Inhaltsverzeichnis

Part I Introduction.- Latin Squares Based on Groups.- When is a Latin Square Based on a Group?.- Part II Admissable Groups.- The Existence Problem for Complete Mappings: The Hall-Paige Conjecture.- Some Classes of Admissible Groups.- The Groups GL(<i>n</i>,<i>q</i>), SL(<i>n</i>,<i>q</i>), PGL(<i>n</i>,<i>q</i>), and PSL(<i>n</i>,<i>q</i>).- Minimal Counterexamples to the Hall-Paige Conjecture.- A Proof of the Hall-Paige Conjecture.- Part III Orthomorphism Graphs of Groups.- Orthomorphism Graphs of Groups.- Elementary Abelian Groups I.- Elementary Abelian Groups II.- Extensions of Orthomorphism Graphs.- ω(<i>G</i>) for Some Classes of Nonabelian Groups.- Groups of Small Order.- Part IV Additional Topics.- Projective Planes from Complete Sets of Orthomorphisms.- Related Topics.- Problems.- References.- Index.

Autorenportrait

<b>Anthony B. Evans</b> is Professor of Mathematics at Wright State University in Dayton, Ohio. Since the mid 1980s, his primary research has been on orthomorphisms and complete mappings of finite groups and their applications. These mappings arise in the study of mutually orthogonal latin squares that are derived from the multiplication tables of finite groups. As an offshoot of this research, he has also worked on graph representations. His previous book, <i>Orthomorphism Graphs of Groups</i> (1992), appeared in the series, <b>Lecture Notes in Mathematics</b>.

Back cover copy

<div><div><div>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.  </div><div><br></div><div>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.  </div><div><br></div><div>Expanding the author’s 1992 monograph, <i>Orthomorphism Graphs of Groups</i>, 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. </div></div></div>

Sonderbeitrag

Presents the first unified proof of the Hall–Paige conjecture Discusses the actions of groups on designs derived from latin squares Includes an extensive list of open problems on the construction and structure of orthomorphism graphs suitable for researchers and graduate students