# Group cohomology of Klein four-group

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 given as follows:

- The chain group is a sum of copies of , indexed by ordered pairs where . In other words, the chain group is:

- The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:

- The map is multiplication by zero if is odd and is multiplication by two if is even.
- The map is multiplication by zero if is odd and multiplication by two if is even.

## Homology groups

### Over the integers

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

The first few homology groups are given below:

rank of as an elementary abelian 2-group | -- | 2 | 1 | 3 | 2 | 4 |

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 (see group cohomology of cyclic group:Z2) using the Kunneth formula for group homology. They can also be computed explicitly using the chain complex description above.

Here is the computation using the Kunneth formula for group homology: [SHOW MORE]### Over an abelian group

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

Here, is the quotient of by and .

These homology groups can be computed in terms of the homology groups over integers using the universal coefficients theorem for group homology.

### Important case types for abelian groups

Case on | Conclusion about odd-indexed homology groups, i.e., | Conclusion about even-indexed homology groups, i.e., |
---|---|---|

is uniquely 2-divisible, i.e., every element of has a unique half. This includes the case that is a field of characteristic not 2. | all zero groups | all zero groups |

is 2-torsion-free, i.e., no nonzero element of doubles to zero | ||

is 2-divisible, but not necessarily uniquely so, e.g., | ||

, any natural number | ||

is a finite abelian group | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of |

is a finitely generated abelian group | all isomorphic to where is the rank for the 2-Sylow subgroup of the torsion part of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of | all isomorphic to where is the rank for the 2-Sylow subgroup of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of |

## Cohomology groups for trivial group action

### Over the integers

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

The first few cohomology groups are given below:

0 | ||||||

rank of as an elementary abelian 2-group | -- | 0 | 2 | 1 | 3 | 2 |

### Over an abelian group

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

Here, is the quotient of by and .

These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.

The first few groups are given below:

### Important case types for abelian groups

Case on | Conclusion about odd-indexed cohomology groups, i.e., | Conclusion about even-indexed homology groups, i.e., |
---|---|---|

is uniquely 2-divisible, i.e., every element of has a unique half. This includes the case that is a field of characteristic not 2. | all zero groups | all zero groups |

is 2-torsion-free, i.e., no nonzero element of doubles to zero | ||

is 2-divisible, but not necessarily uniquely so, e.g., | ||

, any natural number | ||

is a finite abelian group | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of |

is a finitely generated abelian group | all isomorphic to where is the rank for the 2-Sylow subgroup of the torsion part of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of | all isomorphic to where is the rank for the 2-Sylow subgroup of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of |

## Cohomology ring

### Over the integers

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

### Over a 2-divisible ring

If is a 2-divisible unital ring, then it is also a *uniquely* 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and is isomorphic to , occurring in the .

In particular, this includes the case a field of characteristic not 2, as well as a ring (not necessarily a field) of finite positive characteristic.

### Over characteristic two

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

## Second cohomology groups and extensions

### Schur multiplier and Schur covering groups

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

There are two possibilities for the Schur covering group: dihedral group:D8 and quaternion group. These belong to the Hall-Senior family (up to isoclinism). They are precisely the stem extensions where the acting group is the Klein four-group and the base group is its Schur multiplier, namely cyclic group:Z2. For more, see second cohomology group for trivial group action of V4 on Z2.

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