# Difference between revisions of "Group cohomology of Klein four-group"

(→Second cohomology groups and extensions) |
(→Cohomology ring with coefficients in integers) |
||

Line 25: | Line 25: | ||

{{fillin}} | {{fillin}} | ||

− | ==Cohomology ring | + | ==Cohomology groups and cohomology ring== |

− | {{ | + | ===Groups over the integers=== |

+ | |||

+ | The cohomology groups with coefficients in the integers are given as below: | ||

+ | |||

+ | <math>H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = </math> | ||

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

## Revision as of 15:17, 21 July 2011

This article gives specific information, namely, group cohomology, about a particular group, namely: Klein four-group.

View group cohomology of particular groups | View other specific information about Klein four-group

## Classifying space and corresponding chain complex

The classifying space of the Klein four-group is the product space , where is infinite-dimensional real projective space.

A chain complex that can be used to compute the homology of this space is

## Homology groups

### Over the integers

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

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 using the Kunneth formula.

### Over an abelian group

The homology groups with coefficients in an abelian group (which may be equipped with additional structure as a module over a ring ) are given as follows:

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

## Cohomology groups and cohomology ring

### Groups over the integers

The cohomology groups with coefficients in the integers are given as below:

## Second cohomology groups and extensions

### Schur multiplier

The Schur multiplier, defined as the second cohomoogy group for trivial group action and also as the second homology group , is isomorphic to cyclic group:Z2.

See also the projective representation theory of Klein four-group.