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

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The homology groups over the integers are given as follows: | The homology groups over the integers are given as follows: | ||

− | <math>H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & <math>(\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 3 \pmod 4 \\ q \equiv 1 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \equiv 2 \pmod 4 \mbox{ even }, q > 0 \\ \mathbb{Z}, & q = 0 \\\end{array}</math> | + | <math>H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & <math>(\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 3 \pmod 4 \\ q \equiv 1 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \equiv 2 \pmod 4 \mbox{ even }, q > 0 \\ \mathbb{Z}, & q = 0 \\\end{array}\right.</math> |

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

## Revision as of 04:50, 15 January 2013

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 over the integers are given as follows:

**Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & <math>(\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 3 \pmod 4 \\ q \equiv 1 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \equiv 2 \pmod 4 \mbox{ even }, q > 0 \\ \mathbb{Z}, & q = 0 \\\end{array}\right.**

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.

### Schur covering groups

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 (as an abstract group) | Order of second cohomology group | Extensions | Number of extensions up to pseudo-congruence, i.e., number or orbits under automorphism group actions | Cohomology information |
---|---|---|---|---|---|---|---|

cyclic group:Z2 | 2 | 1 | elementary abelian group:E8 | 8 | 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 | 6 | second cohomology group for trivial group action of D8 on Z2 |

cyclic group:Z4 | 4 | 1 | elementary abelian group:E8 | 8 | direct product of D8 and Z4, nontrivial semidirect product of Z4 and Z8, SmallGroup(32,5), central product of D16 and Z4, SmallGroup(32,15), wreath product of Z4 and Z2 | 6 | second cohomology group for trivial group action of D8 on Z4 |

Klein four-group | 4 | 2 | elementary abelian group:E64 | 64 | [SHOW MORE] | 11 | second cohomology group for trivial group action of D8 on V4 |

### Baer invariants

Subvariety of the variety of groups | General name of Baer invariant | Value of Baer invariant for this group |
---|---|---|

abelian groups | Schur multiplier | cyclic group:Z2 |

groups of nilpotency class at most two | 2-nilpotent multiplier | cyclic group:Z2 |

groups of nilpotency class at most three | 3-nilpotent multiplier | trivial group |

any variety of groups containing all groups of nilpotency class at most three | -- | trivial group |