Cohomology Theory of Finite Groups

  • Kenneth S. Brown
Part of the Graduate Texts in Mathematics book series (GTM, volume 87)


Homology and cohomology are usually thought of as dual to one another. We have seen in Chapter III, for example, that homology has a number of formal properties and that cohomology has “dual” properties. If G is finite, however, then homology and cohomology seem to have similar properties rather than dual ones. For example, since every subgroup H of a finite group G has finite index, we have restriction and corestriction maps for arbitrary subgroups, in both homology and cohomology. For another example, the distinction between induced modules and co-induced modules disappears, so we have a single class I of G-modules (namely, the induced modules ℤGA) with the following properties: (a) Every MI is acyclic for both homology and cohomology. (b) For every G-module M there is a module I such that M is a quotient of and M can be embedded in .


Exact Sequence Finite Group Complete Resolution Short Exact Sequence Finite Type 
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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Kenneth S. Brown
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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