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

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

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Case on <math>M</math> !! Conclusion about odd-indexed homology groups, i.e., <math>H_q, q = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H_q, q = 2,4,6,\dots</math> | ||

+ | |- | ||

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

+ | |- | ||

+ | | <math>M</math> is <math>p</math>-torsion-free, i.e., no nonzero element of <math>M</math> multiplies by <math>p</math> to give zero. || <math>(M/pM)^{(q+3)/2}</math> || <math>(M/pM)^{q/2}</math> | ||

+ | |- | ||

+ | | <math>M</math> is <math>p</math>-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(p))^{(q-1)/2}</math> || <math>(\operatorname{Ann}_M(p))^{(q+2)/2}</math> | ||

+ | |- | ||

+ | | <math>M = \mathbb{Z}/p^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup of the torsion part of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math> || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + sq/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math> | ||

+ | |} | ||

==Cohomology groups for trivial group action== | ==Cohomology groups for trivial group action== | ||

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

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Case on <math>M</math> !! Conclusion about odd-indexed cohomology groups, i.e., <math>H^q, q = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H^q, q = 2,4,6,\dots</math> | ||

+ | |- | ||

+ | | <math>M</math> is uniquely <math>p</math>-divisible, i.e., every element of <math>M</math> can be divided by <matH>p</math> uniquely. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups | ||

+ | |- | ||

+ | | <math>M</math> is <math>p</math>-torsion-free, i.e., no nonzero element of <math>M</math> multiplies by <math>p</math> to give zero. || <math>(M/pM)^{(q-3)/2}</math> || <math>(M/pM)^{(q+2)/2}</math> | ||

+ | |- | ||

+ | | <math>M</math> is <math>p</math>-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(p))^{(q+3)/2}</math> || <math>(\operatorname{Ann}_M(p))^{q/2}</math> | ||

+ | |- | ||

+ | | <math>M = \mathbb{Z}/p^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q - 1)/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup of the torsion part of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math> || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math> | ||

+ | |} |

## Revision as of 23:56, 12 October 2011

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

## 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

### 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 |