522[1]

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5. Linear Shift Registers—Introduction, finite state machines, shift registers, characteristic polynomials and periodicity, randomness, security. 6. Non-linear Algorithms—Introduction, intractability and NP-completeness, shift registers with non-linear feedback, some examples using more than one register. 7. Some Block Cipher Systems—Introduction, some examples of cipher feedback systems, block cipher systems, applications of a block cipher. 8. Applying Cipher Systems—Introduction, key structure, key management, Example 8.1: a strategic asynchronous telegraph system, Example 8.2: a portable tactical on/ofT line system, Example 8.3: an on-line electronic fund transfer system. 9. Speech Security Systems—Introduction, basic concepts, Fourier analysis, some properties of speech, voice message transmission, voice scrambling, analogue to digital (A/D) converters, use of A/D and D/A converters. 10. Public Key Cryptography—Introduction, formal definition, a system due to Merkle and Hellman, the RSA system, another system, authentication. The initiated will know that I am a tyro in these matters; I thought a good test of the quality of the writing would be to give Chapter 1 a fairly detailed reading, and skim over the rest. I found Chapter 1 fascinating. The reader is carried along and familiarised with all the various terms and concepts of the subject before rigorous definitions are given in a later chapter. It is very nicely done, with a lacing of nontrivial but attackable exercises, and a complete example of cryptanalysis to end the chapter. The text is visually pleasant, there are lots of good diagrams, even some photographs; the style is good and even witty at times—very refreshing. JAMES WIEGOLD

GROUP THEORY I (A Series of Comprehensive Studies in Mathematics, 247) By

MICHIO SUZUKI:

pp. 434. DM.118.-; US$55.00. (Springer-Verlag, Berlin, 1982.)

This is a translation by the author of Volume 1 of his Japanese-language text Gunron, published by Iwanami Shoten. The aim of the two volumes is to present an exposition of the theory of finite simple groups as it was in the mid-seventies (starting from scratch, a remarkably tall order). In the preface to the English edition, Professor Suzuki points to the enormous progress in the classification problem, the zenith coming in 1980; unfortunately for the group-theoretical public, he does not feel that the time is yet ripe for a third volume dealing with these advances. Some changes have been introduced into the English version to accommodate the new status of the theory. The present volume consists of three chapters on fairly general group theory, as can be seen from the table of contents: 1. Basic Concepts—The definition of a group and some examples; subgroups; cosets; normal subgroups, factor groups; homomorphisms, isomorphism theorems; automorphisms; permutation groups, G-sets; operator groups, semidirect products; general linear groups. 2. Fundamental Theorems—Theorems about p-groups; theorems of Sylow; subnormal series, Schreier's refinement theorem; the Krull-Remak-Schmidt

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theorem; fundamental theorems on abelian groups; generators and relations; extensions of groups and cohomology theory; applications of cohomology theory, the Schur-Zassenhaus theorem; central extensions, Schur's multiplier; wreath products. 3. Some Special Classes of Groups—Torsion-free abelian groups; symmetric groups and alternating groups; geometry of linear groups; Coxeter groups; surveys of finite simple groups; finite subgroups of two-dimensional special linear groups. The author begins by stating quite clearly what is needed for an understanding of his book: the general theory of fields, Zorn's lemma, linear algebra and the like. The exposition starts at a leisurely pace with the definition and most elementary properties of groups. It is done so nicely that I cast away my doubt as to the appropriateness of such simple material in a book of this sort: it will repay reading by dedicated beginners. The pace soon hots up, and Chapters 2 and 3 contain a great deal of material, with back-up in the form of numerous examples, historical remarks and a host of exercises. Indeed, this is yet another among the by now large number of texts that working group theorists, young and old, will want to have on their shelves—and I do not doubt that the same will go for Volume 2, when the English version arrives. The English is good, with only the rare linguistic oddity (which it is churlish to mention!). JAMES WIEGOLD

REPRESENTATIONS OF REAL REDUCTIVE LIE GROUPS (Progress in Mathematics, 15) By

DAVID

A.

VOGAN, JR.:

pp.754. $35.00. (Birkhauser, Boston, 1981.)

This book deals with recent developments in the theory of representations of reductive Lie groups largely from an algebraic point of view. Although the origins of this theory are in the study of unitary representations, the methods developed by Harish-Chandra necessitate going outside this class to consider continuous representations on Banach spaces. Additionally, many representations have been constructed which are believed to be unitary, but have not yet been proven so to be; until they are, they must be dealt with by the more general theory. Banach space representations are, of course, of interest in their own right since they arise naturally in many situations (non-unitary induction, for example). Let G be a real reductive Lie group, K a maximal compact subgroup, and g the complexified Lie algebra of G. A continuous representation of G is called admissible if its restriction to K has finite multiplicities. The space of smooth /C-finite vectors of an admissible representation forms a representation of both K and g, called the associated Harish-Chandra module. If the original representation is unitary and irreducible, it is a theorem of Harish-Chandra that it is admissible, and determined up to unitary equivalence by its associated Harish-Chandra module. This passage from representations of G to the associated Harish-Chandra modules opens up the way to study the group representations by algebraic methods. The book begins with a preliminary Chapter 0 describing the precise class of groups to be dealt with, together with the correspondence between admissible representations and Harish-Chandra modules. The chapter ends with a description