# Group cohomology of symmetric group:S3

This article gives specific information, namely, group cohomology, about a particular group, namely: symmetric group:S3.

View group cohomology of particular groups | View other specific information about symmetric group:S3

## Classifying space and corresponding chain complex

The homology and cohomology groups are the same as the respective homology and cohomology groups of the configuration space of three unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real vector space on the Topology Wiki.

## Family contexts

Family name | Parameter value | General discussion of group cohomology of family |
---|---|---|

symmetric group of degree | degree | group cohomology of symmetric groups |

dihedral group of degree and order | order , degree | group cohomology of dihedral groups |

## 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

The homology groups with coefficients in the ring of integers are as follows:

### Over an abelian group

The homology groups with coefficients in an abelian group are as follows:

Here, denotes the 2-torsion subgroup of and denotes the 6-torsion subgroup of .

## Cohomology groups for trivial group action

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

First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

### Over the integers

The cohomology groups with coefficients in the ring of integers are as follows:

### Over an abelian group

The cohomology groups with coefficients in an abelian group are as follows:

Here denotes the 2-torsion subgroup of and denotes the 6-torsion subgroup of .