# Group cohomology of elementary abelian group:E8

This article gives specific information, namely, group cohomology, about a particular group, namely: elementary abelian group:E8.

View group cohomology of particular groups | View other specific information about elementary abelian group:E8

## Family contexts

Family | Parameter values | Information on group cohomology of family |
---|---|---|

elementary abelian group of order for a prime | group cohomology of elementary abelian groups | |

elementary abelian group of prime-cube order for a prime | group cohomology of elementary abelian group of prime-cube order |

## 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:

The first few homology groups are given as follows:

( | |||||||||

rank of as an elementary abelian -group | -- | 3 | 3 | 7 | 8 | 13 | 15 | 21 | 24 |

### Over an abelian group

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

Here, is the quotient of by the subgroup and .

### Important case types for abelian groups

Case on | Conclusion for odd-indexed homology groups , | Conclusion for even-indexed homology groups , |
---|---|---|

is uniquely -divisible, i.e., every element of can be uniquely divided by . This includes the case that is a field of characteristic not . | all zero groups | all zero groups |

is -torsion-free, i.e., no nonzero element of multiplies by to give zero. | ||

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

, any natural number | ||

is a finite abelian group | All isomorphic to where is the rank of the -Sylow subgroup of | where is the rank of the 2-Sylow subgroup of |

is a finitely generated abelian group | All isomorphic to where is the rank of the 2-Sylow subgroup of the torsion part of and is the rank of the free part of . | All isomorphic to where is the rank of the 2-Sylow subgroup of the torsion part of and is the rank of the free part of . |

## 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 integers are as follows:

The first few cohomology groups are given below:

0 | |||||||||

rank of as an elementary abelian -group | -- | 0 | 3 | 3 | 7 | 8 | 13 | 15 | 21 |

### Over an abelian group

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

Here, is the quotient of by the subgroup and .

### Important case types for abelian groups

Case on | Conclusion for odd-indexed cohomology groups , | Conclusion for even-indexed cohomology groups , |
---|---|---|

is uniquely -divisible, i.e., every element of can be uniquely divided by . This includes the case that is a field of characteristic not . | all zero groups | all zero groups |

is -torsion-free, i.e., no nonzero element of multiplies by to give zero. | ||

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

, any natural number | ||

is a finite abelian group | All isomorphic to where is the rank of the 2-Sylow subgroup of | where is the rank of the 2-Sylow subgroup of |

is a finitely generated abelian group | All isomorphic to where is the rank of the 2-Sylow subgroup of the torsion part of and is the rank of the free part of . |

## Tate cohomology groups for trivial group action

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## Growth of ranks of cohomology groups

### Over the integers

With the exception of the zeroth homology group and cohomology group, the homology groups and cohomology groups over the integers are all elementary abelian -groups.

For homology groups, the rank (i.e., dimension as a vector space over field:F2) is a function of that is a sum of a quadratic function and a periodic function with period 2. The same is true for the cohomology groups, although the precise description of the periodic function differs.

- For homology groups, choosing the periodic function so as to have mean zero, we get that the quadratic function is , and the periodic function is .
- For cohomology groups, choosing the periodic function so as to have mean zero, we get that the quadratic function is , and the periodic function is .

Note that:

- The cohomology groups grow slightly slower than the homology groups. Basically, the cohomology group is the homology group, so the cohomology groups lag behind slightly.
- The periodic function for the cohomology groups is opposite that for the homology groups. This follows from the dual universal coefficients theorem for group cohomology.

### Over the prime field

If we take coefficients in the prime field field:F2, the ranks of the homology groups and cohomology groups both grow as quadratic functions of . The quadratic function in *both* cases is . Note that in this case, the homology groups and cohomology groups are both vector spaces over , and the cohomology group is the vector space dual of the homology group.

Note that there is no periodic part when we are working over the prime field.