# Group cohomology of special linear group of degree two over a finite field

This article gives specific information, namely, group cohomology, about a family of groups, namely: special linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.

View group cohomology of group families | View other specific information about special linear group of degree two | View other specific information about group families for rings of the type finite field

This article describes the group cohomology of the special linear group of degree two over a finite field. The order (size) of the field is , and the characteristic prime is . is a power of . We denote the group as or .

Each is a finite group with periodic cohomology, and the period is 4. We can confirm this from the subgroup structure of special linear group of degree two over a finite field: all the Sylow subgroups are either cyclic groups or generalized quaternion groups.

The cases (see group cohomology of symmetric group:S3) and (see group cohomology of special linear group:SL(2,3)) are somewhat anomalous.

## Particular cases

(field size) | (underlying prime, field characteristic) | Case for | Group | Order of the group () | Group cohomology page |
---|---|---|---|---|---|

2 | 2 | even | symmetric group:S3 | 6 | group cohomology of symmetric group:S3 |

3 | 3 | odd | special linear group:SL(2,3) | 24 | group cohomology of special linear group:SL(2,3) |

4 | 2 | even | alternating group:A5 | 60 | group cohomology of alternating group:A5 |

5 | 5 | odd | special linear group:SL(2,5) | 120 | group cohomology of special linear group:SL(2,5) |

7 | 7 | odd | special linear group:SL(2,7) | 336 | group cohomology of special linear group:SL(2,7) |

8 | 2 | even | projective special linear group:PSL(2,8) | 504 | group cohomology of projective special linear group:PSL(2,8) |

9 | 3 | odd | special linear group:SL(2,9) | 720 | group cohomology of special linear group:SL(2,9) |

## 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

For any the group form a periodic sequence, i.e., the group is a finite group with periodic cohomology. Note that is anomalous.

Congruence class of mod 4 | for | for |
---|---|---|

1 | ||

2 | 0 | 0 |

3 | ||

0 | 0 | 0 |

For , . exhibits interesting behavior.