# Group cohomology of free abelian groups

From Groupprops

This article gives specific information, namely, group cohomology, about a family of groups, namely: free abelian group.

View group cohomology of group families | View other specific information about free abelian group

This article describes the homology and cohomology groups of the free abelian group with generators.

## Classifying space and corresponding chain complex

The free abelian group has classifying space equal to the -fold torus, i.e., the space . A chain complex that can be used to compute the homology for the classifying space and hence also the group is below:

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## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

Here, denotes the binomial coefficient: the number of subsets of size in a set of size . In particular, the homology group is for .

Note that all homology groups themselves are free abelian groups. The homology groups for small values of and are given below. Note that missing cells correspond to zero homology groups:

### Over an abelian group

The homology groups with coefficients in an arbitrary abelian group are given below: