Group cohomology of elementary abelian groups

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

Value of $n$ Elementary abelian group of order $p^n$ Information on group cohomology Case $p = 2$ Case $p = 3$
0 trivial group group cohomology of trivial group group cohomology of trivial group group cohomology of trivial group
1 group of prime order covered as part of group cohomology of finite cyclic groups group cohomology of cyclic group:Z2 group cohomology of cyclic group:Z3
2 elementary abelian group of prime-square order group cohomology of elementary abelian group of prime-square order group cohomology of Klein four-group group cohomology of elementary abelian group:E9
3 elementary abelian group of prime-cube order group cohomology of elementary abelian group of prime-cube order group cohomology of elementary abelian group:E8 group cohomology of elementary abelian group:E27
4 elementary abelian group of prime-fourth order group cohomology of elementary abelian group of prime-fourth order group cohomology of elementary abelian group:E16 group cohomology of elementary abelian group:E81

• $p$ 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 $H^q(E_{p^n};\_)$.

Homology groups

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

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.
Value of $n$ Polynomial that gives rank of $H_q(E_{p^n};\mathbb{Z})$ for odd positive $n$ Polynomial that gives rank of $H_q(E_{p^n};\mathbb{Z})$ for even positive $n$ Degree of the polynomials (equals $n - 1$) Average of leading coefficients for even and odd (equals $1/(2(n - 1)!)$; note that both leading coefficients are equal to this for $n > 1$) Constant term for polynomial for odd degree, which equals that polynomial minus the polynomial for even degree (equals $2 - (1/2^{n-1})$) Description of polynomial for odd $q$ in terms of $u = (q + 1)/2$ Description of polynomial for even $q$ in terms of binomial polynomials of $u = q/2$
1 1 0 0 1/2 1 1 0
2 $\frac{q + 3}{2}$ $\frac{q}{2}$ 1 1/2 3/2 $\binom{u}{0} + \binom{u}{1}$ $\binom{u}{1}$
3 $\frac{q^2 + 4q + 7}{4}$ $\frac{q^2 + 4q}{4}$ 2 1/4 7/4 $\binom{u}{0} + 2\binom{u}{1} + 2\binom{u}{2}$ $3\binom{u}{1} + 2\binom{u}{2}$
4 $\frac{2q^3 + 15q^2 + 34q + 45}{24}$ $\frac{2q^3 + 15q^2 + 34q}{24}$ 3 1/12 15/8 $\binom{u}{0} + 3\binom{u}{1} + 7\binom{u}{2} + 4\binom{u}{3}$ $6\binom{u}{1} + 9\binom{u}{2} + 4\binom{u}{3}$

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:

Value of $q$ Polynomial in $n$ that gives rank of $H_q(E_{p^n};\mathbb{Z})$ Degree (equals $n$) Leading coefficient (equals $1/(n!)$) Description in terms of binomial polynomials
1 $n$ 1 1 $\binom{n}{1}$
2 $\frac{n(n - 1)}{2}$ 2 1/2 $\binom{n}{2}$
3 $\frac{n(n^2 + 5)}{6}$ 3 1/6 $\binom{n}{1} + \binom{n}{2} + \binom{n}{3}$

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 $q$.

$n$ (rows), $q$ (columns) Polynomial for odd $q$ for fixed $n$ Polynomial for even $q$ for fixed $n$ Rank for $q = 1$ Rank for $q = 2$ Rank for $q = 3$ Rank for $q = 4$ Rank for $q = 5$
Polynomial in $n$ for fixed $q$ -- -- $n$ $\frac{n(n-1)}{2}$ $\frac{n(n^2 + 5)}{6}$  ?  ?
Case $n = 1$ 1 0 1 0 1 0 1
Case $n = 2$ $\frac{q + 3}{2}$ $\frac{q}{2}$ 2 1 3 2 4
Case $n = 3$ $\frac{q^2 + 4q + 7}{4}$ $\frac{q^2 + 4q}{4}$ 3 3 7 8 13
Case $n = 4$ $\frac{2q^3 + 15q^2 + 34q + 45}{24}$ $\frac{2q^3 + 15q^2 + 34q}{24}$ 4 6 14 21 35