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

(→Over an abelian group) |
(→Over an abelian group) |
||

Line 57: | Line 57: | ||

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

+ | |||

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

+ | |||

+ | ===Important case types for abelian groups=== | ||

+ | |||

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

+ | ! Case on <math>R</math> or <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 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}</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-1)/2}</math> || <math>(\operatorname{Ann}_M(2))^{(p+2)/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 + 3)/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) + sp/2}</math> where <math>r</math> is the rank for the 2-Sylow subgroup of <math>M</math> | ||

+ | |} | ||

==Cohomology groups and cohomology ring== | ==Cohomology groups and cohomology ring== |

## Revision as of 22:33, 12 October 2011

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 homomology groups are given below:

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 cohomology groups can be computed in terms of the cohomology groups over integers using the universal coefficients theorem for group homology.

### Important case types for abelian groups

Case on or | 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 |

## Cohomology groups and cohomology ring

### Groups over the integers

The cohomology groups with coefficients in the integers are given as below:

## Second cohomology groups and extensions

### Schur multiplier

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.

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