# Group cohomology of dihedral groups

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

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

We consider here the dihedral group of order and degree (i.e., its natural action is on a set of size ).

## Particular cases

Note that the case is anomalous.

(degree) | (order) | Group | Cohomology information |
---|---|---|---|

2 | 4 | Klein four-group | group cohomology of Klein four-group |

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

4 | 8 | dihedral group:D8 | group cohomology of dihedral group:D8 |

5 | 10 | dihedral group:10 | group cohomology of dihedral group:D10 |

## Homology groups for trivial group action

The below applies to . For , see group cohomology of Klein four-group.

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 odd degree

The homology groups with coefficients in the ring of integers are as follows when the degree is odd:

Note that a dihedral group of this sort is a finite group with periodic cohomology, which fits in with the alternative criterion for having periodic cohomology: every abelian subgroup is cyclic. In this case, the period on the homology groups with integer coefficients is 4.

### Over the integers for even degree

The homology groups with coefficients in the ring of integers are as follows when the degree is even:

### Over an abelian group for odd degree

The homology groups with coefficients in an abelian group are as follows when the degree is odd:

Here, denotes the 2-torsion subgroup of and denotes the -torsion subgroup of .

## Cohomology groups for trivial group action

FACTS TO CHECK AGAINST(cohomology group for trivial group action):

First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

### Over the integers for odd degree

The cohomology groups with coefficients in the ring of integers are as follows when the degree is odd:

### Over an abelian group for odd degree

The cohomology groups with coefficients in an abelian group are as follows when the degree is odd:

Here denotes the 2-torsion subgroup of and denotes the -torsion subgroup of .