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

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===Over the integers=== | ===Over the integers=== | ||

+ | The first few homology groups are given below: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! <math>q<math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> | ||

+ | |- | ||

+ | | <math>H_q</math> || <math>\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z}</math> || ? || ? || ? || ? || ? | ||

+ | |} | ||

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

+ | The first few homology groups with coefficients in an abelian group <math>M</math> are given below: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! <math>q<math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> | ||

+ | |- | ||

+ | | <math>H_q</math> || <math>M</math> || <math>M/2M \oplus M/2M</math> || <math>M/2M \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)</math> || ? || ? || ? || ? || ? | ||

+ | |} | ||

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

Line 17: | Line 31: | ||

===Over the integers=== | ===Over the integers=== | ||

+ | |||

+ | The first few cohomology groups are given below: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! <math>q<math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> | ||

+ | |- | ||

+ | | <math>H^q</math> || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z}</math> || ? || ? || ? || ? | ||

+ | |} | ||

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

+ | |||

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

+ | |||

+ | {| class="sortable" border="1" | ||

+ | |||

+ | |- | ||

+ | | <math>H^q</math> || <math>M</math> || <math>\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2) || <math>M/2M \oplus M/2M \oplus \operatorname{Ann}_M(2)</math> || ? || ? || ? || ? || ? | ||

+ | |} | ||

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

Line 25: | Line 55: | ||

==Second cohomology groups and extensions== | ==Second cohomology groups and extensions== | ||

+ | |||

+ | ===Schur multiplier=== | ||

+ | |||

+ | The [[Schur multiplier]], defined as [[second cohomology group for trivial group action]] <math>H^2(G;\mathbb{C}^\ast)</math>, and also as the second homology group <math>H_2(G;\mathbb{Z})</math>, is [[cyclic group:Z2]]. | ||

+ | |||

+ | This has implications for [[projective representation theory of dihedral group:D8]]. | ||

+ | |||

+ | The three possible [[Schur covering group]]s for [[dihedral group:D8]] are: [[dihedral group:D16]], [[semidihedral group:SD16]], and [[generalized quaternion group:Q16]]. For more, see [[second cohomology group for trivial group action of D8 on Z2]], where these correspond precisely to the [[stem extension]]s. | ||

===Second cohomology groups for trivial group action=== | ===Second cohomology groups for trivial group action=== |

## Revision as of 04:10, 1 November 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 first few homology groups are given below:

? | ? | ? | ? | ? |

### 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

The first few cohomology groups are given below:

0 | ? | ? | ? | ? |

### Over an abelian group

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

? | ? | ? | ? | ? |

## 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 cyclic group:Z2.

This has implications for projective representation theory of dihedral group:D8.

The three possible Schur covering groups for dihedral group:D8 are: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. For more, see second cohomology group for trivial group action of D8 on Z2, where these correspond precisely to the stem 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 |