# Group cohomology of symmetric groups

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

View group cohomology of group families | View other specific information about symmetric group

## Particular cases

(order of symmetric group) | Symmetric group of degree | Group cohomology page | |
---|---|---|---|

1 | 1 | trivial group | group cohomology of trivial group |

2 | 2 | cyclic group:Z2 | group cohomology of cyclic group:Z2 |

3 | 6 | symmetric group:S3 | group cohomology of symmetric group:S3 |

4 | 24 | symmetric group:S4 | group cohomology of symmetric group:S4 |

5 | 120 | symmetric group:S5 | group cohomology of symmetric group:S5 |

6 | 720 | symmetric group:S6 | group cohomology of symmetric group:S6 |

## Classifying space and corresponding chain complex

The homology and cohomology groups of the symmetric group are the same as the respective homology and cohomology groups of the configuration space of 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 projective space on the Topospaces wiki.

## 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, "0" for a group is shorthand for the trivial group. is shorthand for the finite cyclic group .

The homology groups eventually stabilize in the following sense: for any fixed , there exists a large enough (explicit expression for in terms of -- around double?) such that for all . The corresponding stable value of homology group is termed the *stable homology group* of degree for the symmetric groups.

symmetric group of degree | if a finite group with periodic cohomology, period of sequence of homology groups for positive degrees | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | trivial group | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | trivial group | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 2 | cyclic group:Z2 | 2 | 0 | 0 | 0 | 0 | 0 | |||||

3 | 6 | symmetric group:S3 | 4 | 0 | 0 | 0 | 0 | 0 | |||||

4 | 24 | symmetric group:S4 | -- | ||||||||||

5 | 120 | symmetric group:S5 | -- | ||||||||||

6 | 720 | symmetric group:S6 | -- | ||||||||||

7 | 5040 | symmetric group:S7 | -- | ||||||||||

Stable |
-- | -- | -- | ? | ? | ? | ? | ? | ? | ? |