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

From Groupprops

(→Over the integers) |
(→Cohomology groups for trivial group action) |
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===Over an abelian group=== | ===Over an abelian group=== | ||

+ | ===Over an abelian group=== | ||

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+ | The homology groups with coefficients in an abelian group <math>M</math> are as follows: | ||

+ | |||

+ | <math>H_p(D_8;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann}_2(M) & \qquad p \equiv 2 \pmod 4 \\ M/8M, & \qquad p \equiv 3 \pmod 4\\ \operatorname{Ann}_8(M), & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.</math> | ||

==Cohomology groups for trivial group action== | ==Cohomology groups for trivial group action== | ||

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

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+ | <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 2 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & p \ \operatorname{odd} \\\end{array}\right.</math> | ||

+ | |||

+ | ===Over an abelian group=== | ||

+ | |||

+ | The cohomology groups with coefficients in an abelian group <math>M</math> are as follows: | ||

+ | |||

+ | <math>H^p(D_8;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ \operatorname{Ann}_2(M), & \qquad p \equiv 1 \pmod 4 \\ M/2M & \qquad p \equiv 2 \pmod 4 \\ \operatorname{Ann}_8(M), & \qquad p \equiv 3 \pmod 4\\ M/8M, & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.</math> | ||

+ | |||

+ | Here <math>\operatorname{Ann}_2(M)</math> denotes the 2-torsion subgroup of <math>M</math> and <math>\operatorname{Ann}_8(M)</math> denotes the 8-torsion subgroup of <math>M</math>. | ||

==Cohomology ring with coefficients in integers== | ==Cohomology ring with coefficients in integers== |

## Revision as of 01:44, 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:

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

### Over an abelian group

The homology groups with coefficients in an abelian group are as follows:

## 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:

### Over an abelian group

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

Here denotes the 2-torsion subgroup of and denotes the 8-torsion subgroup of .

## 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 |