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## Group cohomology of Klein four-group

, 22:53, 12 October 2011
Cohomology groups for trivial group action
| rank of $H^p$ as an elementary abelian 2-group || -- || 0 || 2 || 1 || 3 || 2
|}

===Over an abelian group===

The cohomology groups with coefficients in an abelian group $M$ are given as follows:

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

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

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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