# Difference between revisions of "Group cohomology of symmetric group:S3"

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

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− | | [[symmetric group]] <math>S_n</math> of degree <math>n</math> || <math>n = 3</math> || [[group cohomology of symmetric groups]] | + | | [[symmetric group]] <math>S_n</math> of degree <math>n</math> || degree <math>n = 3</math> || [[group cohomology of symmetric groups]] |

|- | |- | ||

| [[dihedral group]] <math>D_{2n}</math> of degree <math>n</math> and order <math>2n</math> || order <math>2n = 6</math>, degree <math>n = 3</math> || [[group cohomology of dihedral groups]] | | [[dihedral group]] <math>D_{2n}</math> of degree <math>n</math> and order <math>2n</math> || order <math>2n = 6</math>, degree <math>n = 3</math> || [[group cohomology of dihedral groups]] |

## Latest revision as of 04:09, 15 January 2013

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 .