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Over the integers
{{group family-specific information|
group family = elementary abelian group of prime-square order|
information type = group cohomology|
connective = of}}
 
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>.
===Over the integers===
The homology groups below can be computed using the homology groups for the [[group of prime order]] (see [[group cohomology of finite cyclic groups]]) and combining it with the [[Kunneth formula for group homology]]. <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}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.</math> The even and odd cases can be combined giving the following alternative description: <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/2 + 3(1 - (-1)^q)/4}, & \qquad q > 0 \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.</math>
The first few homology groups are given below:
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>.
| <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==
<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 odd and even cases can be combined as follows:
 
<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/2 + 1/4 + 3(-1)^q/4}, & q > 0\\ \mathbb{Z}, & q = 0 \\\end{array}\right.</math>
The first few cohomology groups are given below:
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===
===Important case types for abelian groups===
| <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-31)/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</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>
|}
 
==Tate cohomology groups for trivial group action==
 
{{fillin}}
 
==Growth of ranks of cohomology groups==
 
===Over the integers===
 
With the exception of the zeroth homology group and cohomology group, the homology groups and cohomology groups over the integers are all elementary abelian <math>p</math>-groups.
 
For the homology groups, the rank (i.e., dimension as a vector space over the field of <math>p</math> elements) is a function of <math>q</math> that is a sum of a linear function (of slope 1/2) and a periodic function (of period 2). The same is true for the cohomology groups, although the precise description of the periodic function differs.
 
* For homology groups, choosing the periodic function so as to have mean zero, we get that the linear function is <math>q \mapsto q/2 + 3/4</math> and the periodic function is <math>3(-1)^{q+1}/4</math>.
* For cohomology groups, choosing the periodic function so as to have mean zero, we get that the linear function is <math>q \mapsto q/2 + 1/4</math> and the periodic function is <math>3(-1)^q/4</math>.
 
Note that:
 
* The intercept for the cohomology groups is 1/4, as opposed to the intercept of 3/4 for the homology groups. This is explained by the somewhat slower start of cohomology groups on account of <math>H^1</math> being torsion-free.
* The periodic parts for homology groups and cohomology groups are negatives of each other, indicating an opposing pattern that is explained by looking at the [[dual universal coefficients theorem for group cohomology]].
 
===Over the prime field===
 
If we take coefficients in the prime field <math>\mathbb{F}_p</math>, then the ranks of the homology and cohomology groups both grow as linear functions of <math>q</math>. The linear function in ''both'' cases is <math>q \mapsto q + 1</math>. Note that in this case, the homology groups and cohomology groups are vector spaces over <math>\mathbb{F}_p</math> and the cohomology group is the vector space dual of the homology group.
 
Note that there is no periodic part when we are working over the prime field.
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