# Group cohomology of alternating group:A5

From Groupprops

This article gives specific information, namely, group cohomology, about a particular group, namely: alternating group:A5.

View group cohomology of particular groups | View other specific information about alternating group:A5

## Family contexts

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

alternating group | 5 | group cohomology of alternating groups |

projective general linear group of degree two over a finite field of size | , i.e., field:F4, so the group is | group cohomology of projective general linear group of degree two over a finite field |

projective special linear group of degree two over a finite field of size | , i.e., field:F5, so the group is | group cohomology of projective special linear group of degree two over a finite field |

## 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(AlternatingGroup(5),1); [ ]

This says that .

#### Second homology group

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

gap> GroupHomology(AlternatingGroup(5),2); [ 2 ]

This says that .

#### First few homology groups

gap> List([1..12],i -> [i,GroupHomology(AlternatingGroup(5),i)]); [ [ 1, [ ] ], [ 2, [ 2 ] ], [ 3, [ 2, 3, 5 ] ], [ 4, [ ] ], [ 5, [ 2, 2 ] [ 6, [ 2 ] ], [ 7, [ 2, 3, 5 ] ], [ 8, [ 2, 2 ] ], [ 9, [ 2, 2 ] ], [ 10, [ 2 ] ], [ 11, [ 2, 2, 2, 3, 5 ] ], [ 12, [ 2, 2 ] ] ]