# Difference between revisions of "Group cohomology of direct product of Z4 and Z2"

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{{cohomology groups for trivial group action facts to check against}} | {{cohomology groups for trivial group action facts to check against}} | ||

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

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

+ | |||

+ | The first few cohomology groups are given below (these are the same as the first few homology groups, shifted over by one): | ||

+ | |||

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

+ | ! <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> || 0 || <math>\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> | ||

+ | |} | ||

+ | |||

+ | ===Over an abelian group=== | ||

+ | |||

+ | Below are the cohomology groups with coefficients in an abelian group <math>M</math>: | ||

+ | |||

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

+ | |||

+ | Here, <math>M/2M</math> represents the quotient of <math>M</math> by the subgroup <math>2M = \{ 2x \mid x \in M \}</math>, <math>M/4M</math> represents the quotient of <math>M</math> by the subgroup <math>4M = \{ 4x \mid x \in M \}</math>, <math>\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}</math> and <math>\operatorname{Ann}_M(4) = \{x \in M \mid 4x = 0 \}</math>. | ||

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

## Latest revision as of 00:57, 25 October 2011

This article gives specific information, namely, group cohomology, about a particular group, namely: direct product of Z4 and Z2.

View group cohomology of particular groups | View other specific information about direct product of Z4 and Z2

## 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:

### Over an abelian group

Here, represents the quotient of by the subgroup , represents the quotient of by the subgroup , and .

## 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 first few cohomology groups are given below (these are the same as the first few homology groups, shifted over by one):

0 |

### Over an abelian group

Below are the cohomology groups with coefficients in an abelian group :

Here, represents the quotient of by the subgroup , represents the quotient of by the subgroup , and .