# Group cohomology of special linear group:SL(2,3)

This article gives specific information, namely, group cohomology, about a particular group, namely: special linear group:SL(2,3).

View group cohomology of particular groups | View other specific information about special linear group:SL(2,3)

This article describes the group cohomology of special linear group:SL(2,3).

## Family contexts

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

double cover of alternating group | degree , i.e., the group | group cohomology of double cover of alternating group |

special linear group of degree two over a finite field of size | , i.e., field:F3, i.e., the group is | group cohomology of special 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 are as follows:

The homology groups have a period of 4, which is in keeping with the fact that is a finite group with periodic cohomology.

## 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(SL(2,3),1); [ 3 ]

This says that .

#### Second homology group

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

gap> GroupHomology(SL(2,3),2); [ ]

This says that .

#### First few homology groups

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