Group cohomology of elementary abelian group of prime-cube order

Suppose $$p$$ is a prime number. We are interested in the elementary abelian group of prime-cube order

$$E_{p^3} = (\mathbb{Z}/p\mathbb{Z})^3 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$$

Over the integers
The homology groups over the integers are given as follows:

$$\! H_q(E_{p^3};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{\frac{q^2 + 4q + 7}{4}}, & \qquad q = 1,3,5,\dots \\ (\mathbb{Z}/p\mathbb{Z})^{\frac{q^2+4q}{4}}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.$$

The first few homology groups are given as follows:

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

$$\! H_q(E_{p^3};M) = \left\lbrace\begin{array}{rl} (M/pM)^{\frac{q^2 + 4q + 7}{4}} \oplus (\operatorname{Ann}_M(p))^{\frac{q^2 + 2q - 3}{4}}, & \qquad q = 1,3,5,\dots \\ (M/pM)^{\frac{q^2 + 4q}{4}} \oplus (\operatorname{Ann}_M(p))^{\frac{q^2 + 2q + 4}{4}}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0\\\end{array}\right.$$

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

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

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

The first few cohomology groups are given below:

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

$$H^q(E_{p^3};M) = \left\lbrace \begin{array}{rl} (M/pM)^{\frac{q^2 + 2q - 3}{4}} \oplus (\operatorname{Ann}_M(p))^{\frac{q^2 + 4q + 7}{4}}, & \qquad q = 1,3,5,\dots \\ (M/pM)^{\frac{q^2 + 2q + 4}{4}} \oplus (\operatorname{Ann}_M(p))^{\frac{q^2 + 4q}{4}}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0 \\\end{array}\right.$$

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

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 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 quadratic function and a periodic function with 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 quadratic function is $$q \mapsto (q^2 + 4q)/4 + (7/8)$$, and the periodic function is $$7(-1)^{q + 1}/8$$.
 * For cohomology groups, choosing the periodic function so as to have mean zero, we get that the quadratic function is $$q \mapsto (q^2 + 2q)/4 + (1/8)$$, and the periodic function is $$7(-1)^q/8$$.

Note that:


 * The cohomology groups grow slightly slower than the homology groups. Basically, the $$q^{tH}$$ cohomology group is the $$(q-1)^{th}$$ homology group, so the cohomology groups lag behind slightly.
 * The periodic function for the cohomology groups is opposite that for the homology groups. This follows from the dual universal coefficients theorem for group cohomology.

Over the prime field
If we take coefficients in the prime field $$\mathbb{F}_p$$, the ranks of the homology groups and cohomology groups both grow as quadratic functions of $$q$$. The quadratic function in both cases is $$q \mapsto (q + 1)(q + 2)/2$$. Note that in this case, the homology groups and cohomology groups are both 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.