Difference sets. Connecting algebra, combinatorics, and geometry (Q2838083)

This is one among the beautiful books on the subject of difference sets that I came across in the field of mathematics and especially in combinatorics because of its lucid style and simplicity. NEWLINENEWLINENEWLINE NEWLINE`Difference sets' as a unique subject intermingles with group theory, combinatorics and combinatorial structures in association with abstract algebra, algebraic number theory, finite geometries, and representation theory. From these, the concepts of group characters and multiplies also arose. NEWLINENEWLINENEWLINE NEWLINEThese interrelated mathematical concepts are highly useful to the scholars and students for their research at undergraduate, graduate and doctoral levels. NEWLINENEWLINENEWLINE NEWLINEThe present book overviews these subjects if not exhaustively but impressively with required theorems sometimes with full proofs and sometimes with comprehensive explanations and required examples. By the study of this book, one gains an opportunity to further explore the subject with confidence in different angles enriching one's vision for further research with the orientation of applications in the real-life situations as the authors mention such lines as well. NEWLINENEWLINENEWLINE NEWLINEA group acting on a set or a structure, for example a symmetric design, contains a difference set, which is interesting because of some technical applications for example in communications or in transportation networks. NEWLINENEWLINENEWLINE NEWLINEA difference set is an additive finite group \(G\) and a non-empty proper subset \(D\) of \(G\) is a \((v, k, \lambda)\)-difference set if \(|G|=v\), \(|D|=k\) and there is an integer \(\lambda\) such that each non-identity element of \(G\) can be expressed in exactly \(\lambda\) ways as a difference \(d_1-d_2\) of elements of \(D\). Equivalently, this requires that each non-identity group element appears \(\lambda\) times in the multi set \(\Delta=\{d_1-d_2 \mid d_1, d_2\in D, d_1 \neq d_2\}\). NEWLINENEWLINENEWLINE NEWLINEThe concept of difference set first appeared in the year 1938 in the paper by \textit{J. Singer} [Trans. Am. Math. Soc. 43, 377--385 (1938; Zbl 0019.00502; JFM 64.0972.04)], entitled ``A theorem in finite projective geometry and some applications to number theory.'' NEWLINENEWLINENEWLINE NEWLINEUsing the group and difference set concepts, a combinatorial design can be constructed. Thus, this book projects numerous mathematical concepts and their usage and applications in diversity and richness as well. NEWLINENEWLINENEWLINE NEWLINEDesign theory is a special area utilized in statistical design of experiments besides coding theory, finite group theory, finite geometries like projective and affine geometries and different and distinct algebraic and combinatorial structures. Designs are incidence structures defined as: Let \(t\) be a non-negative integer. A \(t\)-design is an incidence structure \(D= (P,B)\) in which {\parindent=6mmNEWLINE\begin{itemize}\item[i)]Each block contains \(k\) points and NEWLINE\item[ii)]Each subset of \(t\) points is completely contained in exactly \(\lambda\) blocks for some \(\lambda\geq1\). NEWLINENEWLINENEWLINE\end{itemize}}NEWLINE Whereas group structures arise as sets of functions mapping a set \(X\) into itself. They are invertible transformations \(V \to V\) for some vector space \(V\).NEWLINENEWLINENEWLINEThe specific case of a group is automorphism of a symmetric design. And thus, difference sets bridge group theory and design theory. NEWLINENEWLINENEWLINE NEWLINEThe authors expose the famous Bruck-Ryser-Chowla theorem in an impressive way with numerous examples and exercises. In fact, the proof requires the incidence matrix \(X\) of the design or quadratic forms. The authors use matrix algebra for these purposes. Number theoretic conditions are needed for the existence of a symmetric design. NEWLINENEWLINENEWLINE NEWLINEA multiplier for a difference \(D \subset G\) is an automorphism of \(G\) that maps \(D \to D\), which was introduced by \textit{M. Hall jun.} [in: Combinatorics. Part 3: Combinatorial group theory. Proceedings of the Advanced Study Institute on Combinatorics held at Nijenrode Castle, Breukelen, The Netherlands, July 8--20, 1974. Amsterdam: Mathematisch Centrum. 1--26 (1974; Zbl 0317.05001); Proc. Am. Math. Soc. 7, 975--986 (1957; Zbl 0077.05202)] in cyclic difference sets and was dealt with by the authors. The existence of \((v, k, \lambda)\) symmetric designs is interrelated with the existence of \((v, k, \lambda)\) affine and projective geometries and ultimately with the existence of \((v, k, \lambda)\) difference sets. The authors deal with these concepts widely, Hadamard difference sets are the best example for this. NEWLINENEWLINENEWLINE NEWLINEThe authors expose the lines of applying representation theory to the study of difference sets by extending group representation. They shift from a complex representation row \(\rho\) of \(G\) and the transformation \(\rho(g)\) to a new function \(\chi\rho : G \to C\) called character of row given by the trace: \(\chi\rho(g) = \operatorname{Tr}(\rho(g))\) for \(g\in G\) and further deal with this theory of characters of a finite group \(G\). NEWLINENEWLINENEWLINE NEWLINEThese difference sets are mathematically rich, and they also have many applications to real world problems. Pingzhi Fan studied and developed the theory of sequences. The binary sequences are used in imaging with coded masks, error-correcting codes and so on by using these binary sequences. NEWLINENEWLINENEWLINE NEWLINEThis book lays a good foundation for the study of difference sets together with the subjects related to it and prepares the students for further extensive research.