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Orthogonal Latin Squares Based on Groups

Author: Anthony B. Evans

Publisher:

Published: 2018

Total Pages: 537

ISBN-13: 9783319944319

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.--

Orthogonal Latin Squares Based on Groups

Author: Anthony B. Evans

Publisher: Springer

Published: 2018-08-17

Total Pages: 537

ISBN-13: 3319944304

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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.

Latin Squares and Their Applications

Author: A. Donald Keedwell

Publisher: Elsevier

Published: 2015-07-28

Total Pages: 443

ISBN-13: 0444635580

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Latin Squares and Their Applications, Second edition offers a long-awaited update and reissue of this seminal account of the subject. The revision retains foundational, original material from the frequently-cited 1974 volume but is completely updated throughout. As with the earlier version, the author hopes to take the reader ‘from the beginnings of the subject to the frontiers of research’. By omitting a few topics which are no longer of current interest, the book expands upon active and emerging areas. Also, the present state of knowledge regarding the 73 then-unsolved problems given at the end of the first edition is discussed and commented upon. In addition, a number of new unsolved problems are proposed. Using an engaging narrative style, this book provides thorough coverage of most parts of the subject, one of the oldest of all discrete mathematical structures and still one of the most relevant. However, in consequence of the huge expansion of the subject in the past 40 years, some topics have had to be omitted in order to keep the book of a reasonable length. Latin squares, or sets of mutually orthogonal latin squares (MOLS), encode the incidence structure of finite geometries; they prescribe the order in which to apply the different treatments in designing an experiment in order to permit effective statistical analysis of the results; they produce optimal density error-correcting codes; they encapsulate the structure of finite groups and of more general algebraic objects known as quasigroups. As regards more recreational aspects of the subject, latin squares provide the most effective and efficient designs for many kinds of games tournaments and they are the templates for Sudoku puzzles. Also, they provide a number of ways of constructing magic squares, both simple magic squares and also ones with additional properties. Retains the organization and updated foundational material from the original edition Explores current and emerging research topics Includes the original 73 ‘Unsolved Problems’ with the current state of knowledge regarding them, as well as new Unsolved Problems for further study

Latin Squares

Author: József Dénes

Publisher: Elsevier

Published: 1991-01-24

Total Pages: 469

ISBN-13: 0080867863

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In 1974 the editors of the present volume published a well-received book entitled ``Latin Squares and their Applications''. It included a list of 73 unsolved problems of which about 20 have been completely solved in the intervening period and about 10 more have been partially solved. The present work comprises six contributed chapters and also six further chapters written by the editors themselves. As well as discussing the advances which have been made in the subject matter of most of the chapters of the earlier book, this new book contains one chapter which deals with a subject (r-orthogonal latin squares) which did not exist when the earlier book was written. The success of the former book is shown by the two or three hundred published papers which deal with questions raised by it.

Discrete Mathematics Using Latin Squares

Author: Charles F. Laywine

Publisher: John Wiley & Sons

Published: 1998-09-17

Total Pages: 336

ISBN-13: 9780471240648

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Over the past two decades, research in the theory of Latin Squares has been growing at a fast pace, and new significant developments have taken place. This book offers a unique approach to various areas of discrete mathematics through the use of Latin Squares.

Orthomorphism Graphs of Groups

Author: Anthony B. Evans

Publisher: Springer

Published: 2006-11-15

Total Pages: 120

ISBN-13: 3540475419

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This book is about orthomorphisms and complete mappings of groups, and related constructions of orthogonal latin squares. It brings together, for the first time in book form, many of the results in this area. The aim of this book is to lay the foundations for a theory of orthomorphism graphsof groups, and to encourage research in this area. To this end, many directions for future research are suggested. The material in this book should be accessible to any graduate student who has taken courses in algebra (group theory and field theory). It will mainly be useful in research on combinatorial design theory, group theory and field theory.

The Genius of Euler: Reflections on his Life and Work

Author: William Dunham

Publisher: American Mathematical Soc.

Published: 2020-08-03

Total Pages: 326

ISBN-13: 147045744X

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Combinatorial Methods in Discrete Mathematics

Author: Vladimir N. Sachkov

Publisher: Cambridge University Press

Published: 1996-01-11

Total Pages: 324

ISBN-13: 0521455138

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This is an attempt to present some complex problems of discrete mathematics in a simple and unified form using a unique, general combinatorial scheme. The author's aim is not always to present the most general results, but rather to focus attention on ones that illustrate the methods described. A distinctive aspect of the book is the large number of asymptotic formulae derived.This is an important book, describing many ideas not previously available in English; the author has taken the chance to update the text and references where appropriate.

Combinatorics and Finite Geometry

Author: Steven T. Dougherty

Publisher: Springer Nature

Published: 2020-10-30

Total Pages: 374

ISBN-13: 3030563952

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This undergraduate textbook is suitable for introductory classes in combinatorics and related topics. The book covers a wide range of both pure and applied combinatorics, beginning with the very basics of enumeration and then going on to Latin squares, graphs and designs. The latter topic is closely related to finite geometry, which is developed in parallel. Applications to probability theory, algebra, coding theory, cryptology and combinatorial game theory comprise the later chapters. Throughout the book, examples and exercises illustrate the material, and the interrelations between the various topics is emphasized. Readers looking to take first steps toward the study of combinatorics, finite geometry, design theory, coding theory, or cryptology will find this book valuable. Essentially self-contained, there are very few prerequisites aside from some mathematical maturity, and the little algebra required is covered in the text. The book is also a valuable resource for anyone interested in discrete mathematics as it ties together a wide variety of topics.

Group Theory - Proceedings Of The Biennial Ohio State - Denison Conference

Author: Ronald Solomon

Publisher: World Scientific

Published: 1993-09-30

Total Pages: 350

ISBN-13: 9814553034

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This volume is a collection of invited papers on the theory of groups, most of which were presented at the biennial Ohio State-Denison Conference, May 1992, in memory of Hans Zassenhaus. These papers treat important topics in the theory of p-groups, solvable groups, finitely presented groups, arithmetic groups, monodromy groups and the general structure and representation theory of groups. Of particular note are papers by John Walter on root systems, by Leonard Scott on integral equivalence of permutation representations and Alex Turull on generalized Brauer groups.