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Group cohomology of Klein four-group

790 bytes added, 22:53, 12 October 2011
Cohomology groups for trivial group action
| rank of <math>H^p</math> as an elementary abelian 2-group || -- || 0 || 2 || 1 || 3 || 2
|}
 
===Over an abelian group===
 
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})^{(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>.
 
These can be deduced from the homology groups with coefficients in the integers using the [[dual universal coefficients theorem]].
 
===Important case types for abelian groups===
 
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==Cohomology ring==
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