# Difference between revisions of "Group cohomology of dihedral group:D8"

From Groupprops

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The homology groups with coefficients in the ring of integers are as follows: | The homology groups with coefficients in the ring of integers are as follows: | ||

− | <math>\! H_p(D_8;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even}\end{array}\right | + | <math>\! H_p(D_8;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even}\end{array}\right</math> |

As a sequence (Starting <math>p = 0</math>), the first few homology groups are: | As a sequence (Starting <math>p = 0</math>), the first few homology groups are: |

## Revision as of 01:27, 9 October 2011

This article gives specific information, namely, group cohomology, about a particular group, namely: dihedral group:D8.

View group cohomology of particular groups | View other specific information about dihedral group:D8

## 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 with coefficients in the ring of integers are as follows:

**Failed to parse (syntax error): \! H_p(D_8;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even}\end{array}\right**

As a sequence (Starting ), the first few homology groups are:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

0 | 0 | 0 | 0 |

### Over an abelian group

## 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 cohomology groups with coefficients in the ring of integers are as follows:

## Cohomology ring with coefficients in integers

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

## Second cohomology groups and extensions

### 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 | elementary abelian group:E8 | direct product of D8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 | second cohomology group for trivial group action of D8 on Z2 |

cyclic group:Z4 | 4 | 1 | ? | ? | second cohomology group for trivial group action of D8 on Z4 |