# Group cohomology of elementary abelian group of prime-square order

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

## Contents

## Particular cases

## 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.

### 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 -divisible, i.e., every element of can be divided uniquely 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 | isomorphic to where is the rank (i.e., minimum number of generators) for the -Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the -Sylow subgroup of |

is a finitely generated abelian group | all isomorphic to where is the rank for the -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 -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 -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.

### 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 -divisible, i.e., every element of can be divided by uniquely. This includes the case that is a field of characteristic not 2. | 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 | isomorphic to where is the rank (i.e., minimum number of generators) for the -Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the -Sylow subgroup of |

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