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Group cohomology of Klein four-group

2,315 bytes added, 22:33, 12 October 2011
Over an abelian group
<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==
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