Group cohomology of elementary abelian group of prime-square order

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

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.

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

The even and odd cases can be combined giving the following alternative description:

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

The first few homology groups are given below:

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

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

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

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

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

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

The odd and even cases can be combined as follows:

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

The first few cohomology groups are given below:

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

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

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

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

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 $$p$$-groups.

For the homology groups, the rank (i.e., dimension as a vector space over the field of $$p$$ elements) is a function of $$q$$ 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 $$q \mapsto q/2 + 3/4$$ and the periodic function is $$3(-1)^{q+1}/4$$.
 * For cohomology groups, choosing the periodic function so as to have mean zero, we get that the linear function is $$q \mapsto q/2 + 1/4$$ and the periodic function is $$3(-1)^q/4$$.

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 $$H^1$$ 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 $$\mathbb{F}_p$$, then the ranks of the homology and cohomology groups both grow as linear functions of $$q$$. The linear function in both cases is $$q \mapsto q + 1$$. Note that in this case, the homology groups and cohomology groups are vector spaces over $$\mathbb{F}_p$$ 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.