Introduction Ⅰ.Group Extensions， Simple Algebras and Cohomology Ⅰ.0 Introduction Ⅰ.1 Group Extensions Ⅰ.2 Extensions Associated to the Quaternions The Group of Unit Quaternions and SO（3） The Generalized Quaternion Groups and Binary Tetrahedral Group Ⅰ.3 Central Extensions and S1 Bundles on the Torus T2 Ⅰ.4 The Pull—back Construction and Extensions Ⅰ.5 The Obstruction to Extension when the Center is Non—Trivial The Dependence of／z（gl， g2， g3） on f' and the Lifting L Ⅰ.6 Counting the Number of Extensions Ⅰ.7 The Relation Satisfied by／z（gl， g2， g3） A Certain Universal Extension Each Element in H3φ（G； C） Represents an Obstruction Ⅰ.8 Associative Algebras and H2φ（G； C） Basic Structure Theorems for Central Simple F—Algebras Tensor Products of Central Simple F—Algebras The Cohomological Interpretation of Central Simple Division Algebras Comparing Different Maximal Subfields， the Brauer Group Ⅱ.Classifying Spaces and Group Cohomology Ⅱ.0 Introduction Ⅱ.1 Preliminaries on Classifying Spaces Ⅱ.2 Eilenberg—MacLane Spaces and the Steenrod Algebra at（p） Axioms for the Steenrod Algebra A（2） Axioms for the Steenrod Algebra A（p） The Cohomology of Eilenberg—MacLane Spaces The Hopf Algebra Structure on ，A（p） Ⅱ.3 Group Cohomology Ⅱ.4 Cup Products Ⅱ.5 Restriction and Transfer Transfer and Restriction for Abelian Groups An Alternate Construction of the Transfer Ⅱ.6 The Cartan—Eilenberg Double Coset Formula Ⅱ.7 Tate Cohomology and Applications Ⅱ.8 The First Cohomology Group and Out（G） Ⅲ.Invariants and Cohomology of Groups Ⅲ.0 Introduction Ⅲ.I General Invariants Ⅲ.2 The Dickson Algebra Ⅲ.3 A Theorem of Serre Ⅲ.4 Symmetric Invariants Ⅲ.5 The Cardenas—Kuhn Theorem Ⅲ.6 Discussion of Related Topics and Further Results The Dickson Algebras and Topology The Ring of Invariants for Sp2n（F2） The Invariants for Subgroups of GL4（F2） Ⅳ.Spectral Sequences and Detection Theorems Ⅳ.0 Introduction Ⅳ.1 The Lyndon—Hochschild—Serre Spectral Sequence： Geometric Approach Wreath Products
Central Extensions A Lemma of Quillen—Venkov Ⅳ.2 Change of Rings and the Lyndon—Hochschild—Serre Spectral Sequence The Dihedral Group D2n The Quaternion Group Q8 Ⅳ.3 Chain Approximations in Acyclic Complexes Ⅳ.4 Groups with Cohomology Detected by Abelian Subgroups Ⅳ.5 Structure Theorems for the Ring H*（G； Fp） Evens—Venkov Finite Generation Theorem The Quillen—Venkov Theorem The Krull Dimension of H*（G； Fp） Ⅳ.6 The Classification and Cohomology Rings of Periodic Groups The Classification of Periodic Groups The mod（2） Cohomology of the Periodic Groups Ⅳ.7 The Definition and Properties of Steenrod Squares The Squaring Operations The P—Power Operations for p Odd Ⅴ.G.Complexes and Equivariant Cohomology Ⅴ.0 Introduction to Cohomological Methods Ⅴ.1 Restriction on Group Actions Ⅴ.2 General Properties of Posets Associated to Finite Groups Ⅴ.3 Applications to Cohomoiogy The Sporadic Group M11 The Sporadic Group J1 Ⅵ.The Cohomology of the Symmetric Groups Ⅵ.0 Introduction Ⅵ.I Detection Theorems for H*（Sn； Fp） and Construction of Generators The Sylow p—Subgroups of Sn The Conjugacy Classes of Elementary p—Subgroups in Sn Weak Closure Properties for Vn（p）□ Sylp，（Spn） and （Vn—i（p））pi □ Spn—1□Z／p The Image of res *： H*（Spn；IFp）→H*（Vn（p）；Fp） Ⅵ.2 Hopf Algebras The Theorems of Borel and Hopf Ⅵ.3 The Structure of H*（Sn；Fp） Ⅵ.4 More Invariant Theory Ⅵ.5 H*（Sn）， n = 6， 8， 10， 12 Ⅵ.6 The Cohomology of the Alternating Groups Ⅶ.Finite Groups of Lie Type Ⅶ.1 Preliminary Remarks Ⅶ.2 The Classical Groups of Lie Type Ⅶ.3 The Orders of the Finite Orthogonal and Symplectic Groups. Ⅶ.4 The Cohomology of the Groups GLn（q） Ⅶ.5 The Cohomology of the Finite Orthogonal Groups Ⅶ.6 The Groups H*（Sp2n（q）； F2） Ⅶ.7 The Exceptional Chevalley Groups Ⅷ.Cohomology of Sporadic Simple Groups Ⅷ.0 Introduction Ⅷ.1 The Cohomology of M11 Ⅷ.2 The Cohomology of J1 Ⅷ.3 The Cohomology of M12
The Structure of the Mathieu Group M12 Ⅷ.4 Discussion of H*（M12； F2） Ⅷ.5 The Cohomology of Other Sporadic Simple Groups The O'Nan Group O'N The Rank Four Sporadic Groups The Lattice of Subgroups of 2 □ 2 □ 2 The Cohomology Structure of 22+4 Detection and the Cohomology of J2， J3 The Cohomology of the Groups M22， M23， SU4（3）， McL， and Ly Remark on the Cohomology of M23 Ⅸ.The Plus Construction and Applications Ⅸ.0 Preliminaries Ⅸ.1 Definitions Ⅸ.2 Classification and Construction of Acyclic Maps Ⅸ.3 Examples and Applications The Infinite Symmetric Group The General Linear Group over a Finite Field The Binary Icosahedral Group The Mathieu Group M12 The Group J1 The Mathieu Group M23 Ⅸ.4 The Kan—Thurston Theorem Ⅹ.The Schur Subgroup of the Brauer Group Ⅹ.0 Introduction Ⅹ.1 The Brauer Groups of Complete Local Fields Valuations and Completions The Brauer Groups of Complete Fields with Finite Valuations Ⅹ.2 The Brauer Group and the Schur Subgroup for Finite Extensions of Q The Brauer Group of a Finite Extension of Q The Schur Subgroup of the Brauer Group The Group Q／Z and its Aut Group Ⅹ.3 The Explicit Generators of the Schur Subgroup Cyclotomic Algebras and the Brauer—Witt Theorem The Galois Group of the Maximal Cyclotomic Extension of F The Cohomological Reformulation of the Schur Subgroup Ⅹ.4 The Groups H*cont（GF； Q／Z） and H*cont（Gv； Q／Z） The Cohomology Groups H*cont（GF； Q／Z） The Local Cohomology with Q／Z Coefficients The Explicit Form of the Evaluation Maps at the Finite Valuations Ⅹ.5 The Explicit Structure of the Schur Subgroup， S（F） The Map H*cont（Gv；Q／Z）→H2coont（Gv；Qp，cycl） The Invariants at the Infinite Real Primes The Remaining Local Maps References Index
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