Topics in applied abstract algebra (Q2849044)

The aim of the book under review is to present a variety of concrete applications of modern abstract algebra to practical problems in various areas in information theory, communication theory, and natural sciences. As the authors point out in preface, the text is addressed to two categories of readers: (1) those who are familiar with the basics of abstract algebra and want to see some interesting applications of it to real-world problems, and (2) those who have to deal with such applications in their area of science or engineering and are interested in acquiring a profound knowledge of the pure mathematics involved.NEWLINENEWLINE In this regard, the book may be used as a text for a course in applied abstract algebra for senior-level students in various fields. In fact, the abstract algebraic theory used in the applications is sufficiently developed in the course of the text, thereby keeping the book basically self-contained for a wide audience of interested readers.NEWLINENEWLINE Although the book is not meant to serve as a text for a course in abstract algebra per se, it may well be used as a supplementary text for a regular course in abstract algebra as well as a companion volume to all the standard texts in this field. As for the precise contents, the material is organized in seven chapters, each of which is divided into several sections.NEWLINENEWLINENEWLINEChapter 0 gives a condensed review of the basic algebraic structures and their specific properties, including groups and semigroups, their applications in elementary number theory, rings, ideals, and (finite) fields.NEWLINENEWLINEChapter 1 discusses Boolean algebras and their use in the theory of switching circuits, whereas Chapter 2 describes different methods of constructing balanced incomplete block designs via difference sets, quadratic residues, finite fields, and planar near rings, respectively.NEWLINENEWLINEChapter 3 gives an introduction to algebraic cryptography, thereby treating algebraic enciphering algorithms and classical cryptosystems, block ciphers and the advanced encryption standard, the Rijndael algorithm as well as the knapsack cryptosystem and the RSA cryptosystem.NEWLINENEWLINEChapter 4 is devoted to coding theory, with particular emphasis on error-correcting codes, linear codes, cyclic codes, and BCH codes. The necessary mathematical tools from polynomial algebra and number theory are developed along the way.NEWLINENEWLINEChapter 5 illustrates the applications of symmetry groups and Burnside's theorem to the analysis of colorings and color patterns. Further topics of this chapter concern Pólya's theorem and its application in this context, on the one hand, and generating functions for non-isomorphic graphs on the other.NEWLINENEWLINEFinally, Chapter 6 presents the theory of classification of wallpaper pattern groups in the context of symmetry groups of the real plane.NEWLINENEWLINE NEWLINEIt should be pointed out that the single chapters are independent of one another and may be studied in any order. However, the mathematical depth increases from chapter to chapter, where the last two topics are already rather sophisticated and use quite a bit of involved group theory.NEWLINENEWLINE NEWLINEEach section of the book ends with a series of related exercises, and numerous examples throughout the text effectively help the reader grasp the abstract concepts and methods.NEWLINENEWLINE Altogether, this book offers a wealth of interesting applications of various field of abstract algebra, which are presented in a highly lucid, detailed, rigorous and enlightening style of writing. No doubt, this text is a very useful source for students, instructors, and practitioners likewise.