This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Grobner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics."

This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.

Group theory is one of the most fundamental branches of mathematics. This volume of the Encyclopaedia is devoted to two important subjects within group theory. The first part of the book is concerned with infinite groups. The authors deal with combinatorial group theory, free constructions through group actions on trees, algorithmic problems, periodic groups and the Burnside problem, and the structure theory for Abelian, soluble and nilpotent groups. They have included the very latest developments; however, the material is accessible to readers familiar with the basic concepts of algebra. The second part treats the theory of linear groups. It is a genuinely encyclopaedic survey written for non-specialists. The topics covered includethe classical groups, algebraic groups, topological methods, conjugacy theorems, and finite linear groups. This book will be very useful to allmathematicians, physicists and other scientists including graduate students who use group theory in their work.

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

From the reviews: ..". [Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ..." --The Mathematical Gazette

From the reviews: ..". The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ..." Bulletin of the London Mathematical Society, 1996

This textbook, written by a dedicated and successful pedagogue who developed the present undergraduate algebra course at Moscow State University, differs in several respects from other algebra textbooks available in English. The book reflects the Soviet approach to teaching mathematics with its emphasis on applications and problem-solving -- note that the mathematics department in Moscow is called the I~echanics-Mathematics" Faculty. In the first place, Kostrikin's textbook motivates many of the algebraic concepts by practical examples, for instance, the heated plate problem used to introduce linear equations in Chapter 1. In the second place, there are a large number of exercises, so that the student can convert a vague passive understanding to active mastery of the new ideas. Thes~ problems are intended to be challenging but doable by the student; the harder ones have hints at the back of the book. This feature also makes the book ideally suited for learning algebra on one's own outside of the framework of an organized course. In the third place, the author treats material which is usually not part of an elementary course but which is fundamental in applications. Thus, Part II includes an introduction to the classical groups and to representation theory. With many American colleges now trying to bring their undergraduate mathematics curriculum closer to applications, it seems worthwhile to translate Soviet textbooks which reflect their greater experience in this area of mathematical pedagogy.