# Group cohomology of quaternion group

From Groupprops

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

View group cohomology of particular groups | View other specific information about quaternion group

## 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 are given as follows:

The group is a finite group with periodic cohomology, in keeping with the other definition of being a group with periodic cohomology: every abelian subgroup is cyclic.

The first few homology groups are given below:

0 | 0 |

### Over an abelian group

The first few homology groups with coefficients in an abelian group are given below:

? | ? | ? | ? | ? |

## 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

0 | ? | ? | ? | ? |

### Over an abelian group

The first few cohomology groups with coefficients in an abelian group are as follows:

? | ? | ? | ? | ? |

## Cohomology ring with coefficients in integers

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

## Second cohomology groups and extensions

### Schur multiplier

The Schur multiplier, defined as second cohomology group for trivial group action , and also as the second homology group , is the trivial group.

### Schur covering groups

Since the Schur multiplier is a trivial group, the Schur covering group of the quaternion group is the quaternion group itself.

### Second cohomology groups for trivial group action

Group acted upon | Order | Second part of GAP ID | Second cohomology group for trivial group action | Extensions | Cohomology information |
---|---|---|---|---|---|

cyclic group:Z2 | 2 | 1 | Klein four-group | direct product of Q8 and Z2 (1 time), nontrivial semidirect product of Z4 and Z4 (3 times) | second cohomology group for trivial group action of Q8 on Z2 |

cyclic group:Z4 | 4 | 1 | Klein four-group | direct product of Q8 and Z4 (1 time), nontrivial semidirect product of Z4 and Z8 (3 times) | second cohomology group for trivial group action of Q8 on Z4 |

Klein four-group | 4 | 2 | elementary abelian group:E16 | direct product of Q8 and V4 (1 time), SmallGroup(32,2), direct product of SmallGroup(16,4) and Z2 | second cohomology group for trivial group action of Q8 on V4 |