# Group cohomology of groups of order 8

This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of order 8.

View group cohomology of group families | View group cohomology of groups of a particular order |View other specific information about groups of order 8

Group | GAP ID 2nd part | Hall-Senior number | Nilpotency class | If it is a finite group with periodic cohomology, the period for the homology groups | Cohomology information page |
---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 2 | group cohomology of cyclic group:Z8, see also group cohomology of finite cyclic groups |

direct product of Z4 and Z2 | 2 | 2 | 1 | -- | group cohomology of direct product of Z4 and Z2 |

dihedral group:D8 | 3 | 4 | 2 | -- | group cohomology of dihedral group:D8 |

quaternion group | 4 | 5 | 2 | 4 | group cohomology of quaternion group |

elementary abelian group:E8 | 5 | 1 | 1 | -- | group cohomology of elementary abelian group:E8 |

With the exception of the zeroth homology group and cohomology group, *all* homology and cohomology groups over *all* possible abelian groups are 2-groups.

## 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 table below lists the first few homology groups with coefficients in the integers. We use to denote the cyclic group of order .

We use 0 to denote the trivial group.

Group | GAP ID 2nd part | Hall-Senior number | Nilpotency class | (= abelianization) | (= Schur multiplier) | |||
---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 0 | 0 | |||

direct product of Z4 and Z2 | 2 | 2 | 1 | |||||

dihedral group:D8 | 3 | 4 | 2 | |||||

quaternion group | 4 | 5 | 2 | 0 | 0 | |||

elementary abelian group:E8 | 5 | 1 | 1 |

### Over an abelian group

Below are the homology groups for trivial group action with coefficients in an abelian group . We denote by the subgroup and by the subgroup .

These groups can be computed from the homology groups with coefficients in the integers by using the universal coefficients theorem for group homology.

Group | GAP ID 2nd part | Hall-Senior number | Nilpotency class | |||||
---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | |||||

direct product of Z4 and Z2 | 2 | 2 | 1 | |||||

dihedral group:D8 | 3 | 4 | 2 | |||||

quaternion group | 4 | 5 | 2 | |||||

elementary abelian group:E8 | 5 | 1 | 1 |

## 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

The table below lists the first few cohomology groups with coefficients in the integers. We use to denote the cyclic group of order . Note that each cohomology group is just the previous homology group, i.e., . This is a consequence of the dual universal coefficients theorem for group cohomology.

All the groups are trivial groups. We use 0 to denote the trivial group.

Group | GAP ID 2nd part | Hall-Senior number | Nilpotency class | |||||
---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 0 | 0 | 0 | ||

direct product of Z4 and Z2 | 2 | 2 | 1 | 0 | ||||

dihedral group:D8 | 3 | 4 | 2 | 0 | ? | ? | ? | |

quaternion group | 4 | 5 | 2 | 0 | ? | ? | ? | |

elementary abelian group:E8 | 5 | 1 | 1 | 0 |

### Over an abelian group

Group | GAP ID 2nd part | Hall-Senior number | Nilpotency class | |||||
---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | |||||

direct product of Z4 and Z2 | 2 | 2 | 1 | |||||

dihedral group:D8 | 3 | 4 | 2 | |||||

quaternion group | 4 | 5 | 2 | ? | ? | ? | ||

elementary abelian group:E8 | 5 | 1 | 1 |

## Second cohomology groups and extensions

### Schur multiplier and Schur covering groups

FACTS TO CHECK AGAINSTfor Schur multiplier of group of prime power orderABELIAN CASE: Schur multiplier of finite abelian group is its exterior squareUPPER BOUNDS: upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order |upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order and exponent of centerLOWER BOUNDS: lower bound on size of Schur multiplier for group of prime power order based on minimum size of generating set

The Schur multiplier is defined as second cohomology group for trivial group action, , and also as the second homology group . A corresponding Schur covering group of is a group that arises as a stem extension with base normal subgroup the Schur multiplier and the quotient group is .

### 2-nilpotent multipliers

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

### Higher nilpotent multipliers

Note that nilpotent multiplier of nilpotent group for class higher than its class is trivial, so the 3-nilpotent multiplier and higher nilpotent multipliers are all trivial.