Group cohomology of elementary abelian groups

We are interested in describing the homology groups and cohomology groups for an elementary abelian group of order $$p^n$$. This can be viewed as the additive group of a $$n$$-dimensional vector space over a field of $$p$$ elements. It is isomorphic to the external direct product of $$n$$ copies of the group of prime order.

Particular cases
For this article, we will use the following notation:


 * $$p$$ is the underlying prime of the elementary abelian group.
 * $$n$$ is the rank of the elementary abelian group (i.e., its dimension as a vector space over $$\mathbb{F}_p$$).
 * $$q$$ is the degree in which we are looking at the homology or cohomology, i.e., we are looking at $$H_q$$ or $$H^q$$.

The group will be denoted $$E_{p^n}$$ and the homology/cohomology as $$H_q(E_{p^n};\_)$$ or $$H^q(E_{p^n};\_)$$.

Over the integers
All the formulas obtained here are obtained by combining information about the group cohomology of finite cyclic groups with the Kunneth formula for group homology, as well as basic facts about computation of tensor products and $$\operatorname{Ext}$$ for finitely generated abelian groups. Details of the derivations are pending.

Rank as polynomial in homology degree for fixed rank of elementary abelian group
The zeroth homology group is always $$\mathbb{Z}$$. All higher homology groups are elementary abelian $$p$$-groups. For fixed $$n$$, there are two polynomials in $$q$$, both of degree $$n - 1$$ (one for even $$q$$, one for odd $$q$$) such that the rank of $$H_q(E_{p^n};\mathbb{Z})$$ is that polynomial in $$q$$. The polynomials are given below. In all cases, the following are true:


 * Both polynomials have degree $$n - 1$$.
 * For $$n > 1$$, the leading coefficient of both polynomials is $$1/(2(n - 1)!)$$ (is it? Just guesswork right now).
 * The polynomials differ only in their constant terms, with the polynomial for even $$q$$ having zero constant term.

Rank as polynomial in rank for fixed degree of elementary abelian group
The zeroth homology group is always $$\mathbb{Z}$$. All higher homology groups are elementary abelian $$p$$-groups. For fixed $$q$$, we can find a polynomial in $$n$$ such that the rank of $$H_q(E_{p^n};\mathbb{Z})$$ is that polynomial in $$q$$. The polynomials are given below:

Combined information on ranks
Below are the ranks of the homology groups $$H_q(E_{p^n};\mathbb{Z})$$ in terms of both $$n$$ and $$q$$.