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Cohomology of finite groups. 2nd ed. (English) Zbl 1061.20044
Grundlehren der Mathematischen Wissenschaften 309. Berlin: Springer (ISBN 3-540-20283-8/hbk). viii, 324 p. EUR 89.00/net; sFr. 149.50; £ 69.00; \$ 109.00 (2004).

Since its origins, in the 1940s, the cohomology theory of groups has been the common way of interaction between algebra and topology, contributing to the development of many areas in mathematics, like group theory, number theory, homological algebra, representation theory, algebraic K-theory, homotopy theory, group actions, classifying spaces and characteristic classes.

This book is very different from other treatments since it emphasizes the computational aspects of the cohomology of finite groups with coefficients in a field. Unfortunately, computing the cohomology of a finite group can be quite difficult. In the last years various sophisticated methods have been employed, and the book is devoted to the presentation of some of them making explicit computations of numerous examples. Other methods, like Gröbner bases (D. J. Green, Gröbner bases and the computation of group cohomology. [Lect. Notes Math. 1828. Berlin: Springer (2003; Zbl 1050.20036)]), have also been recently used to make calculations on cohomology of groups, but these are not appointed on the book.

Chapter I contains the theory of group extensions, their relationship with low dimensional cohomology of groups and the study of the classification of finite dimensional central simple algebras over a field.

Chapter II is one of the basic chapters of the book. It studies the basic structure of classifying spaces and gives basic techniques to calculate cohomology of groups, particulary, finite groups.

In this 2nd edition, Chapter III on invariants and cohomology of groups has been revised and expanded [for the first ed., 1994, see Zbl 0820.20060]. The role of classical invariant theory for ${𝔽}_{p}G$-modules necessary for the cohomological calculations later in the book is discussed here. They motivate and illustrate in detail the Dickson invariants ($G={\text{GL}}_{n}\left({𝔽}_{p}\right)$) examining a new example in this edition, $p=2$ and $n=3$. This chapter also contains a proof of Serre’s theorem on the product of Bocksteins, and the Cárdenas-Kuhn theorem for weakly closed systems. In the last section of this chapter, results on Dickson algebras and Conway’s sporadic simple group, that did not appear in the first edition, are incorporated.

Chapter IV introduces the main computational techniques: spectral sequences. It discusses the Lyndon-Hochschild-Serre spectral sequence for a group extension, first from a geometric point of view and then from the algebraic point of view following the methods of Liulevicius and Wall. It considers Quillen’s results on the Krull dimension of the cohomology ring, detection theorems for the cohomology of wreath products and they use ideas developed in the chapter to obtain a different construction of the Steenrod operations.

Chapter V exposes the role played by group cohomology in the theory of group actions. They begin with a discussion of the spectral sequence of the Borel construction for equivariant cohomology, and applications to the discussion of exponents in group cohomology. Applying techniques introduced by Quillen and K. Brown to certain simplicial complexes defined from a collection of subgroups of a group, and using results due to K. Brown and Webb, it is shown that the cohomology of the group can be computed using an alternating sum with the cohomology of the stabilizers of either one of the complexes above.

Chapter VI contains an extended exposition of the structure of the homology and cohomology of the symmetric groups, and they use techniques of Hopf algebras, in concrete, apply the Borel-Hopf theorems to obtain the structure of the Hopf algebras ${H}_{*}\left({S}_{\infty };{𝔽}_{p}\right)$. In the last two sections, they give complete calculations of the cohomology of the symmetric and alternating groups with coefficients in ${𝔽}_{2}$ for $n=6,8,10$ and 12.

Chapter VII describes the main features of the classical finite groups of Lie type and calculates the cohomology of the general linear groups over a finite field, finite orthogonal groups, finite symplectic groups and some exceptional Chevalley groups.

Chapter VIII contains complete calculations of the cohomology of some sporadic simple groups: four of the five Mathieu groups ${M}_{11}$, ${M}_{12}$, ${M}_{22}$, ${M}_{23}$, the Janko groups ${J}_{1}$, ${J}_{2}$, ${J}_{3}$, the O’Nan group ${O}^{\text{'}}N$, the McLaughlin group $McL$ and the Lyons group $Ly$. In fact, this chapter is precisely the one in which more new developments can be found with respect to the first edition since the authors incorporate results obtained over ${M}_{22}$, ${M}_{23}$, ${J}_{2}$, ${J}_{3}$, $McL$ and $Ly$ by the authors and their collaborators in the past decade.

Chapter IX introduces acyclic groups, describes Quillen’s plus construction with applications and a proof of the Kan-Thurston theorem which states that the cohomology of any topological space is the cohomology of a group.

The book ends with a chapter where cohomological methods are used to present a solution to the Schur subgroup problem of the Brauer group, which consists in classifying all the division algebras that can occur in the rational group algebra of a finite group.

New references arising from recent developments in the field have been added.

“Cohomology of finite groups” is a book that covers a long range of material of interest to people working in group theory, number theory, homological algebra, representation theory, and other related areas. I say merely that each mathematician interested in algebra and topology should have a copy of this book on their shelf and make sure that their librarian gets one as well. Overall I thoroughly recommend this book and believe that it will be a useful book for introducing students to cohomological methods for groups.

##### MSC:
 20J06 Cohomology of groups 20-02 Research monographs (group theory) 20J05 Homological methods in group theory 55R35 Classifying spaces of groups and $H$-spaces 55R40 Homology of classifying spaces, characteristic classes 18G40 Spectral sequences (homological algebra) 20C20 Modular representations and characters of groups 19A22 Frobenius induction, Burnside and representation rings 20C05 Group rings of finite groups and their modules (group theory) 16K40 Infinite dimensional and general division rings 18G10 Resolutions; derived functors (category theory)