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

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The homology groups with coefficients in an abelian group <math>M</math> are given as follows: | The homology groups with coefficients in an abelian group <math>M</math> are given as follows: | ||

− | <math>H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/pM)^{(q+3)/2} \oplus (\operatorname{Ann}_M(p))^{(q-1)/2}, & \qquad q = 1,3,5,\dots\\ (M/pM)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0 \\\end{array}\right.</math> | + | <math>H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/ pM)^{(q+3)/2} \oplus (\operatorname{Ann}_M(p))^{(q-1)/2}, & \qquad q = 1,3,5,\dots\\ (M/pM)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0 \\\end{array}\right.</math> |

Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>. | Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>. |

## Revision as of 16:06, 24 October 2011

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

## Contents

## Particular cases

## 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 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

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 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

### 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 |