View source for Group cohomology of elementary abelian group of prime-square order

Suppose <math>p</math> is a [[prime number]]. We are interested in the [[elementary abelian group of prime-square order]] <math>E_{p^2} = (\mathbb{Z}/p\mathbb{Z})^2 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}</math>. 

==Particular cases==

{| class="sortable" border="1"
! Value of prime <math>p</math> !! [[elementary abelian group of prime-square order]] !! cohomology information
|-
| 2 || [[Klein four-group]] || [[group cohomology of Klein four-group]]
|-
| 3 || [[elementary abelian group:E9]] || [[group cohomology of elementary abelian group:E9]]
|-
| 5 || [[elementary abelian group:E25]] || [[group cohomology of elementary abelian group:E25]]
|}

==Homology groups for trivial group action==

{{homology groups for trivial group action facts to check against}}

===Over the integers===

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

The first few homology groups are given below:

{| class="sortable" border="1"
! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math>
|-
| <math>H_q</math> || <math>\mathbb{Z}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math> || <math>\mathbb{Z}/p\mathbb{Z}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^3 = E_{p^3}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^4 = E_{p^4}</math>
|-
| rank of <math>H_q</math> as an elementary abelian <math>p</math>-group || -- || 2 || 1 || 3 || 2 || 4
|}

===Over an abelian group===

The homology groups with coefficients in an abelian group <math>M</math> are given as follows:

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

Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>.

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

===Important case types for abelian groups===

{| class="sortable" border="1"
! Case on <math>M</math> !! Conclusion about odd-indexed homology groups, i.e., <math>H_q, q = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H_q, q = 2,4,6,\dots</math>
|-
| <math>M</math> is uniquely <math>p</math>-divisible, i.e., every element of <math>M</math> can be divided uniquely by <math>p</math>. This includes the case that <math>M</math> is a field of characteristic not <math>p</math>. || all zero groups || all zero groups
|-
| <math>M</math> is <math>p</math>-torsion-free, i.e., no nonzero element of <math>M</math> multiplies by <math>p</math> to give zero. || <math>(M/pM)^{(q+3)/2}</math> || <math>(M/pM)^{q/2}</math>
|-
| <math>M</math> is <math>p</math>-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(p))^{(q-1)/2}</math> || <math>(\operatorname{Ann}_M(p))^{(q+2)/2}</math>
|-
| <math>M = \mathbb{Z}/p^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math>
|-
| <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math>
|-
| <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}</math> where <math>r</math> is the rank for the <math>p</math>-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}/p\mathbb{Z})^{r(q + 1) + sq/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup 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>
|}

==Cohomology groups for trivial group action==

{{cohomology groups for trivial group action facts to check against}}

===Over the integers===

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

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

The first few cohomology groups are given below:

{| class="sortable" border="1"
! <math>q</math> !! <math>\! 0 </math> !! <math>\! 1 </math> !! <math>\! 2</math> !! <math>\! 3 </math> !! <math>\! 4 </math> !! <math>\! 5</math>
|-
| <math>H^q</math> || <math>\mathbb{Z}</math> || 0 || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math> || <math>\mathbb{Z}/p\mathbb{Z}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^3 = E_{p^3}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}</math>
|-
| rank of <math>H^q</math> as an elementary abelian <math>p</math>-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^q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace \begin{array}{rl} (\operatorname{Ann}_M(p))^{(q+3)/2} \oplus (M/pM)^{(q-1)/2}, & q = 1,3,5,\dots \\ (\operatorname{Ann}_M(p))^{q/2} \oplus (M/pM)^{(q+2)/2}, & q = 2,4,6,\dots \\ M, & q = 0 \\\end{array}\right.</math>

Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>.

These can be deduced from the homology groups with coefficients in the integers using the [[dual universal coefficients theorem for group cohomology]].

===Important case types for abelian groups===

{| class="sortable" border="1"
! Case on <math>M</math> !! Conclusion about odd-indexed cohomology groups, i.e., <math>H^q, q = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H^q, q = 2,4,6,\dots</math>
|-
| <math>M</math> is uniquely <math>p</math>-divisible, i.e., every element of <math>M</math> can be divided by <matH>p</math> uniquely. 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 <math>p</math>-torsion-free, i.e., no nonzero element of <math>M</math> multiplies by <math>p</math> to give zero. || <math>(M/pM)^{(q-3)/2}</math> || <math>(M/pM)^{(q+2)/2}</math>
|-
| <math>M</math> is <math>p</math>-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(p))^{(q+3)/2}</math> || <math>(\operatorname{Ann}_M(p))^{q/2}</math>
|-
| <math>M = \mathbb{Z}/p^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math> || <math>(\mathbb{Z}/p\mathbb{Z})^{q+1}</math>
|-
| <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the <math>p</math>-Sylow subgroup of <math>M</math>
|-
| <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q - 1)/2}</math> where <math>r</math> is the rank for the <math>p</math>-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}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}</math> where <math>r</math> is the rank for the <math>p</math>-Sylow subgroup 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>
|}