# Difference between revisions of "Group cohomology of Klein four-group"

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

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

|- | |- | ||

| <math>M</math> is uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. 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 uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups | ||

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

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

Here, <math>M/2M</math> is the quotient of <math>M</math> by <math>2M = \{ 2x \mid x \in M \}</math> and <math>\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}</math>. | Here, <math>M/2M</math> is the quotient of <math>M</math> by <math>2M = \{ 2x \mid x \in M \}</math> and <math>\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}</math>. | ||

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

+ | |||

+ | The first few groups are given below: | ||

+ | |||

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

+ | ! <math>p</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> | ||

+ | |- | ||

+ | | <math>H^p</math> || <math>M</math> || <math>(\operatorname{Ann}_M(2))^2</math> || <math>(\operatorname{Ann}_M(2)) \oplus (M/2M)^2</math> || <math>(\operatorname{Ann}_M(2))^3 \oplus (M/2M)</math> || <math>(\operatorname{Ann}_M(2))^2 \oplus (M/2M)^3</math> || <math>(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2</math> | ||

+ | |} | ||

===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^p, p = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H^p, p = 2,4,6,\dots</math> | ||

+ | |- | ||

+ | | <math>M</math> is uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. 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 2-torsion-free, i.e., no nonzero element of <math>M</math> doubles to zero || <math>(M/2M)^{(p-3)/2}</math> || <math>(M/2M)^{(p+2)/2}</math> | ||

+ | |- | ||

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

+ | |- | ||

+ | | <math>M = \mathbb{Z}/2^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math> | ||

+ | |- | ||

+ | | <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p - 1)/2}</math> where <math>r</math> is the rank for the 2-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}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}</math> where <math>r</math> is the rank for the 2-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 ring== | ||

+ | |||

+ | ===Over the integers=== | ||

{{fillin}} | {{fillin}} | ||

− | == | + | ===Over a 2-divisible ring=== |

+ | |||

+ | If <math>R</math> is a 2-divisible unital ring, then it is also a ''uniquely'' 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and <math>H^*(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};R)</math> is isomorphic to <math>R</math>, occurring in the <math>H^0</math>. | ||

+ | |||

+ | In particular, this includes the case <math>R</math> a field of characteristic not 2, as well as <math>R</math> a ring (not necessarily a field) of finite positive characteristic. | ||

+ | |||

+ | ===Over characteristic two=== | ||

+ | |||

+ | {{fillin}} | ||

==Second cohomology groups and extensions== | ==Second cohomology groups and extensions== | ||

− | ===Schur multiplier=== | + | ===Schur multiplier and Schur covering groups=== |

The [[Schur multiplier]], defined as the [[second cohomology group for trivial group action]] <math>H^2(G,\mathbb{C}^\ast)</math> and also as the second homology group <math>H_2(G,\mathbb{Z})</math>, is isomorphic to [[cyclic group:Z2]]. | The [[Schur multiplier]], defined as the [[second cohomology group for trivial group action]] <math>H^2(G,\mathbb{C}^\ast)</math> and also as the second homology group <math>H_2(G,\mathbb{Z})</math>, is isomorphic to [[cyclic group:Z2]]. | ||

+ | |||

+ | There are two possibilities for the [[Schur covering group]]: [[dihedral group:D8]] and [[quaternion group]]. These belong to the Hall-Senior family <math>\Gamma_2</math> (up to [[isoclinism]]). They are precisely the [[stem extension]]s where the acting group is the [[Klein four-group]] and the base group is its Schur multiplier, namely [[cyclic group:Z2]]. For more, see [[second cohomology group for trivial group action of V4 on Z2]]. | ||

See also the [[projective representation theory of Klein four-group]]. | See also the [[projective representation theory of Klein four-group]]. | ||

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===Second cohomology groups for trivial group action=== | ===Second cohomology groups for trivial group action=== | ||

+ | <section begin="second cohomology group for trivial group action summary"/> | ||

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

− | ! Group acted upon !! Order !! Second part of GAP ID !! [[Second cohomology group for trivial group action]] !! Extensions !! Cohomology information | + | ! Group acted upon !! Order !! Second part of GAP ID !! [[Second cohomology group for trivial group action]] (as abstract group) !! Order !! Extensions !! Number of extensions up to [[pseudo-congruent extension|pseudo-congruence]], i.e., number of orbits under action of relevant automorpism groups !! Cohomology information |

|- | |- | ||

− | | [[cyclic group:Z2]] || 2 || 1 || [[elementary abelian group:E8]] || [[elementary abelian group:E8]], [[direct product of Z4 and Z2]], [[quaternion group]] and [[dihedral group:D8]] || [[second cohomology group for trivial group action of V4 on Z2]] | + | | [[cyclic group:Z2]] || 2 || 1 || [[elementary abelian group:E8]] || 8 || [[elementary abelian group:E8]], [[direct product of Z4 and Z2]], [[quaternion group]] and [[dihedral group:D8]] || 4 || [[second cohomology group for trivial group action of V4 on Z2]] |

|- | |- | ||

− | | [[cyclic group:Z4]] || 4 || 1 || [[elementary abelian group:E8]] || [[direct product of Z4 and V4]], [[direct product of Z8 and Z2]], [[central product of D8 and Z4]], [[M16]] || [[second cohomology group for trivial group action of V4 on Z4]] | + | | [[cyclic group:Z4]] || 4 || 1 || [[elementary abelian group:E8]] || 8 || [[direct product of Z4 and V4]], [[direct product of Z8 and Z2]], [[central product of D8 and Z4]], [[M16]] || 4 || [[second cohomology group for trivial group action of V4 on Z4]] |

|- | |- | ||

− | | [[Klein four-group]] || 4 || 2 || [[elementary abelian group:E64]] || [[elementary abelian group:E16]], [[direct product of Z4 and Z4]], [[direct product of Z4 and V4]], [[direct product of D8 and Z2]], [[direct product of Q8 and Z2]], [[SmallGroup(16,3)]], [[nontrivial semidirect product of Z4 and Z4]] || [[second cohomology group for trivial group action of V4 on V4]] | + | | [[Klein four-group]] || 4 || 2 || [[elementary abelian group:E64]] || 64 || [[elementary abelian group:E16]], [[direct product of Z4 and Z4]], [[direct product of Z4 and V4]], [[direct product of D8 and Z2]], [[direct product of Q8 and Z2]], [[SmallGroup(16,3)]], [[nontrivial semidirect product of Z4 and Z4]] || 7 || [[second cohomology group for trivial group action of V4 on V4]] |

|- | |- | ||

− | | [[cyclic group:Z8]] || 8 || 1 || [[elementary abelian group:E8]] || [[direct product of Z8 and V4]], [[direct product of Z16 and Z2]], [[central product of D8 and Z8]], [[M32]] || [[second cohomology group for trivial group action of V4 on Z8]] | + | | [[cyclic group:Z8]] || 8 || 1 || [[elementary abelian group:E8]] || 8 || [[direct product of Z8 and V4]], [[direct product of Z16 and Z2]], [[central product of D8 and Z8]], [[M32]] || 4 || [[second cohomology group for trivial group action of V4 on Z8]] |

|- | |- | ||

− | | [[direct product of Z4 and Z2]] || 8 || 2 || ? || ? || | + | | [[direct product of Z4 and Z2]] || 8 || 2 || [[elementary abelian group:E64]] || 64 || ? || ? || [[second cohomology group for trivial group action of V4 on direct product of Z4 and Z2]] |

|- | |- | ||

− | | [[elementary abelian group:E8]] || 8 || 5 || ? || ? || | + | | [[elementary abelian group:E8]] || 8 || 5 || [[elementary abelian group:E512]] || 512 || ? || ? || [[second cohomology group for trivial group action of V4 on E8]] |

|} | |} | ||

+ | <section end="second cohomology group for trivial group action summary"/> |

## Latest revision as of 03:38, 23 November 2012

This article gives specific information, namely, group cohomology, about a particular group, namely: Klein four-group.

View group cohomology of particular groups | View other specific information about Klein four-group

## Classifying space and corresponding chain complex

The classifying space of the Klein four-group is the product space , where is infinite-dimensional real projective space.

A chain complex that can be used to compute the homology of this space is given as follows:

- The chain group is a sum of copies of , indexed by ordered pairs where . In other words, the chain group is:

- The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:

- The map is multiplication by zero if is odd and is multiplication by two if is even.
- The map is multiplication by zero if is odd and multiplication by two if is even.

## Homology groups

### Over the integers

The homology groups with coefficients in the ring of integers are given as follows:

The first few homology groups are given below:

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

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 (see group cohomology of cyclic group:Z2) using the Kunneth formula for group homology. They can also be computed explicitly using the chain complex description above.

Here is the computation using the Kunneth formula for group homology: [SHOW MORE]### 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 2-divisible, i.e., every element of has a unique half. This includes the case that is a field of characteristic not 2. | all zero groups | all zero groups |

is 2-torsion-free, i.e., no nonzero element of doubles to zero | ||

is 2-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 2-Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of |

is a finitely generated abelian group | all isomorphic to where is the rank for the 2-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 2-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 2-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.

The first few groups are given below:

### 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 2-divisible, i.e., every element of has a unique half. This includes the case that is a field of characteristic not 2. | all zero groups | all zero groups |

is 2-torsion-free, i.e., no nonzero element of doubles to zero | ||

is 2-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 2-Sylow subgroup of | isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of |

is a finitely generated abelian group | all isomorphic to where is the rank for the 2-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 2-Sylow subgroup of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of |

## Cohomology ring

### Over the integers

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Over a 2-divisible ring

If is a 2-divisible unital ring, then it is also a *uniquely* 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and is isomorphic to , occurring in the .

In particular, this includes the case a field of characteristic not 2, as well as a ring (not necessarily a field) of finite positive characteristic.

### Over characteristic two

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Second cohomology groups and extensions

### Schur multiplier and Schur covering groups

The Schur multiplier, defined as the second cohomology group for trivial group action and also as the second homology group , is isomorphic to cyclic group:Z2.

There are two possibilities for the Schur covering group: dihedral group:D8 and quaternion group. These belong to the Hall-Senior family (up to isoclinism). They are precisely the stem extensions where the acting group is the Klein four-group and the base group is its Schur multiplier, namely cyclic group:Z2. For more, see second cohomology group for trivial group action of V4 on Z2.

See also the projective representation theory of Klein four-group.