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The Indian National. Science Academy has planned to bring out monographs on special topics with the aim of providing acce~sible surveys/reviews of topics of current research in various fields. Prof. S. K. Malik, FNA, Editor of Publications INSA asked me in October 1997 to edit a volume on algebra in this series. I invited a number of algebraists, several of them working in group rings, and it is with great satisfaction and sincere thanks to the authors that I present here in Algebra: Some Recent Advances the sixteen contributions received in response to my invitations. I. B. S. Passi On Abelian Difference Sets K. r Arasu and Surinder K. Sehgal 1. Introduction We review some existence and nonexistence results - new and old - on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics. Difference sets are very important in combinatorial design theory and in commu nication engineering while designing sequences with good correlation properties. Our extended bibliography covers a wide variety of papers written in the area of difference sets and related topics.
List of contents
On Abelian Difference Sets.- Unit Groups of Group Rings.- Projective Modules Over Polynomial Rings.- Around Automorphisms of Relatively Free Groups.- Jordan Decomposition.- On the Normalizer Problem.- values at Zero and the Galois Structure of Global Units.- Central Units in Integral Group Rings.- Alternative Loop Rings and Related Topics.- L-values at Zero and the Galois Structure of Global Units.- A Survey of Groups in which Normality or Permutability is a Transitive Relation.- The Structure of Some Group Rings.- Symmetric Elements and Identities in Group Algebras.- Serial Modules and Rings.- On Subgroups Determined by Ideals of an Integral Group Ring.- A Complex Irreducible Representation of the Quaternion Group and a Non-free Projective Module over the Polynomial Ring in Two Variables over the Real Quaternions.
