《群与格引论——有限群与正定有理格》介绍正定有理格及它们有限等距群理论的，可作为代数群论、表示论等方向的低年级研究生教材。对其中的某些主题，给出了作者独到的观点。本书应该是第一本介绍有限群表示论及有理格之间的分界面的书籍。这个分界面是目前数学许多领域感兴趣的主题。作者格里斯在本书介绍的方法大多用一般的形式表达，以便于读者进一步研究。目前涉及与本书相同主题的书是1988年Springer出版的百科全书SPLAG，那是本经典的书籍，但不能作为教材。本书着重基础群论，并不同于传统的经典理论。

1 Introduction

 1.1 Outline of the book

 1.2 Suggestions for further reading

 1.3 Notations, background, conventions

2 Bilinear Forms, Quadratic Forms and Their Isometry Groups

 2.1 Standard results on quadratic forms and reflections

2.1.1 Principal ideal domains （PIDs）

 2.2 Linear algebra

2.2.1 Interpretation of nonsingularity

2.2.2 Extension of scalars

2.2.3 Cyclicity of the values of a rational bilinea.r form

2.2.4 Gram matrix

 2.3 Discriminant group

 2.4 Relations between a lattice and sublattices

 2.5 Involutions on quadratic spaces

 2.6 Standard results on quadratic forms and reflections, II

2.6.1 Involutions on lattices

 2.7 Scaled isometries: norm doublers and triplers

3 General Results on Finite Groups and Invariant Lattices

 3.1 Discreteness of rational lattices

 3.2 Finiteness of the isometry group

 3.3 Construction of a G-invariant bilinear form

 3.4 Semidirect products and wreath products

 3.5 Orthogonal decomposition of lattices

4 Root Lattices of Types A, D, E

 4.1 Background from Lie theory

 4.2 Root lattices, their duals and their isometry groups

4.2.1 Definition of the AN lattices

4.2.2 Definition of the Dn lattices

4.2.3 Definition of the En lattices

4.2.4 Analysis of the A,n root lattices

4.2.5 Analysis of the Dn root lattices

4.2.6 More on the isometry groups of type Dn

4.2.7 Analysis of the En root lattices

5 Hermite and Minkowski Functions

 5.1 Small ranks and small determinants

5.1.1 Table for the Minkowski and Hermite functions

5.1.2 Classifications of small rank, small determinant lattices

 5.2 Uniqueness of the lattices E6, E7 and Es

 5.3 More small ranks and small determinants

6 Constructions of Lattices by Use of Codes

 6.1 Definitions and basic results

6.1.1 A construction of the Es-lattice with the binary [8, 4, 4] cod

6.1.2 A construction of the Es-lattice with the ternary [4, 2, 3] cod

 6.2 The proofs

6.2.1 About power sets, boolean sums and quadratic forms

6.2.2 Uniqueness of the binary [8, 4, 4] code

6.2.3 Reed-Muller codes

6.2.4 Uniqueness of the tetracode

6.2.5 The automorphism group of the tetracode

6.2.6 Another characterization of [8, 4, 4]2

6.2.7 Uniqueness of the Es-lattice implies uniqueness of the binary [8, 4, 4] code

 6.3 Codes over F7 and a （mod 7）-construction of Es

6.3.1 The A6-1attice

7 Group Theory and Representations

 7.1 Finite groups

 7.2 Extraspecial p-groups

7.2.1 Extraspecial groups and central products

7.2.2 A normal form in an extraspecial group

7.2.3 A classification of extraspecial groups

7.2.4 An application to automorphism groups of extraspecial groups

 7.3 Group representations

7.3.1 Representations of extraspecial p-groups

7.3.2 Construction of the BRW groups

7.3.3 Tensor products

 7.4 Representation of the BRW group G

7.4.1 BRW groups as group extensions

8 Overview of the Barnes-Wall Lattices

 8.1 Some properties of the series

 8.2 Commutator density

8.2.1 Equivalence of 2/4-, 3/4-generation and commutator density for Dihs

8.2.2 Extraspecial groups and commutator density

9 Construction and Properties of the Barnes-Wall Lattices

 9.1 The Barnes-Wall series and their minimal vectors

 9.2 Uniqueness for the BW lattices

 9.3 Properties of the BRW groups

 9.4 Applications to coding theory

 9.5 More about minimum vectors

10 Even unimodular lattices in small dimensions

 10.1 Classifications of even unimodular lattices

 10.2 Constructions of some Niemeier lattices

10.2.1 Construction of a Leech lattice

 10.3 Basic theory of the Golay code

10.3.1 Characterization of certain Reed-Muller codes

10.3.2 About the Golay code

10.3.3 The octad Triangle and dodecads

10.3.4 A uniqueness theorem for the Golay code

 10.4 Minimal vectors in the Leech lattice

 10.5 First proof of uniqueness of the Leech lattice

 10.6 Initial results about the Leech lattice

10.6.1 An automorphism which moves the standard frame ...

 10.7 Turyn-style construction of a Leech lattice

 10.8 Equivariant unimodularizations of even lattices

11 Pieces of Eight

 11.1 Leech trios and overlattices

 11.2 The order of the group O（A）

 11.3 The simplicity of M24

 11.4 Sublattices of Leech and subgroups of the isometry group

 11.5 Involutions on the Leech lattice

References

Index

Appendix A The Finite Simple Groups