An Introduction

Group Cohomology

An Introduction

Group cohomology is a powerful tool for studying groups. It is a generalization of ordinary cohomology theory, which is used to study topological spaces. Group cohomology can be used to study the structure of groups, their representations, and their actions on other mathematical objects.

Basic Definitions

The cohomology of a group G with coefficients in an abelian group A is defined as the homology of the complex ``` 0 → A → A ⊗ G → A ⊗ G<sup>2</sup> → ... ``` where the map from A ⊗ Gⁿ to A ⊗ Gⁿ⁺¹ is given by ``` a ⊗ g<sub>1</sub> ⊗ ... ⊗ g<sub>n</sub> ↦ g<sub>1</sub>a ⊗ g<sub>2</sub> ⊗ ... ⊗ g<sub>n</sub> - a ⊗ g<sub>1</sub>g<sub>2</sub> ⊗ ... ⊗ g<sub>n</sub> ``` The cohomology groups of G with coefficients in A are denoted by Hⁿ(G,A).

Resolutions

A resolution of a group G is a chain complex ``` ... → P<sub>n</sub> → ... → P<sub>1</sub> → P<sub>0</sub> → 0 ``` such that the homology of the complex is isomorphic to G. Resolutions are used to compute the cohomology of G.

Low Degree Cohomology

The low degree cohomology groups of a group G are often used to study the structure of G. For example, the first cohomology group H¹(G,A) is isomorphic to the group of homomorphisms from G to A. The second cohomology group H²(G,A) is isomorphic to the group of extensions of G by A.

More General Cohomology Theories

Group cohomology can be generalized to other algebraic structures, such as rings and modules. The cohomology theory of rings is called Hochschild cohomology, and the cohomology theory of modules is called group cohomology with coefficients in a module.