# Difference between revisions of "Group cohomology of elementary abelian group of prime-square order"

(→Over the integers) |
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! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> | ! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> | ||

|- | |- | ||

− | | <math>H_q</math> || <math>\mathbb{Z}</math> || <math>(\mathbb | + | | <math>H_q</math> || <math>\mathbb{Z}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math> || <math>\mathbb{Z}/p\mathbb{Z}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^3 = E_{p^3}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^4 = E_{p^4}</math> |

|- | |- | ||

| rank of <math>H_q</math> as an elementary abelian <math>p</math>-group || -- || 2 || 1 || 3 || 2 || 4 | | rank of <math>H_q</math> as an elementary abelian <math>p</math>-group || -- || 2 || 1 || 3 || 2 || 4 |

## Revision as of 23:48, 12 October 2011

Suppose is a prime number. We are interested in the elementary abelian group of prime-square order .

## Contents

## Homology groups

### Over the integers

The first few homology groups are given below:

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

### 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.

## 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 -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.