# Group cohomology of symmetric group:S4

This article gives specific information, namely, group cohomology, about a particular group, namely: symmetric group:S4.

View group cohomology of particular groups | View other specific information about symmetric group:S4

## 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 four 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 |

projective general linear group of degree two over a finite field of size | , i.e., field:F3 | group cohomology of projective general linear group of degree two over a finite field |

## 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 over the integers have a description in terms of the congruence class of the homology degree modulo 4 as well as its quotient on integer division by 4.

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

The first few homology groups are given as follows:

( |

## GAP implementation

### Computation of homology groups

The homology groups for trivial group action on the integers can be computed using the `Hap` package (if the package is installed but not automatically loaded, load it using `LoadPackage("hap");`), specifically its GroupHomology function. The function returns a list of numbers which are the orders of cyclic groups whose external direct product is the desired homology group.

#### First homology group

The first homology group, which is also the abelianization, can be computed as follows:

gap> GroupHomology(SymmetricGroup(4),1); [ 2 ]

This says that .

#### Second homology group

The second homology group, which is also the Schur multiplier, can be computed as follows:

gap> GroupHomology(SymmetricGroup(4),2); [ 2 ]

This says that .

#### First few homology groups

gap> List([1..10],i -> [i,GroupHomology(SymmetricGroup(4),i)]); [ [ 1, [ 2 ] ], [ 2, [ 2 ] ], [ 3, [ 2, 4, 3 ] ], [ 4, [ 2 ] ], [ 5, [ 2, 2, 2 ] ], [ 6, [ 2, 2 ] ], [ 7, [ 2, 2, 4, 3 ] ], [ 8, [ 2, 2, 2 ] ], [ 9, [ 2, 2, 2, 2 ] ], [ 10, [ 2, 2, 2 ] ] ]