Changes

Group cohomology of Klein four-group

, 22:33, 12 October 2011
Over an abelian group
$H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/2M)^{(p+3)/2} \oplus (\operatorname{Ann}_M(2))^{(p-1)/2}, & \qquad p = 1,3,5,\dots\\ (M/2M)^{p/2} \oplus (\operatorname{Ann}_M(2))^{(p+2)/2}, & \qquad p = 2,4,6,\dots \\ M, & \qquad p = 0 \\\end{array}\right.$

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 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 $R$ or $M$ !! Conclusion about odd-indexed homology groups, i.e., $H_p, p = 1,3,5,\dots$!! Conclusion about even-indexed homology groups, i.e., $H_p, p = 2,4,6,\dots$
|-
| $M$ is uniquely 2-divisible, i.e., every element of $M$ has a unique half. This includes the case that $M$ is a field of characteristic not 2. || all zero groups || all zero groups
|-
| $M$ is 2-torsion-free, i.e., no nonzero element of $M$ doubles to zero || $(M/2M)^{(p+3)/2}$ || $(M/2M)^{p/2}$
|-
| $M$ is 2-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$ || $(\operatorname{Ann}_M(2))^{(p-1)/2}$ || $(\operatorname{Ann}_M(2))^{(p+2)/2}$
|-
| $M = \mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number || $\mathbb{Z}/2\mathbb{Z})^{p+1}$ || $(\mathbb{Z}/2\mathbb{Z})^{p+1}$
|-
| $M$ is a finite abelian group || isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$ || isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$
|-
| $M$ is a finitely generated abelian group || all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$ || all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + sp/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$
|}
==Cohomology groups and cohomology ring==