# Group cohomology of elementary abelian group of prime-fourth order

## Contents

This article gives specific information, namely, group cohomology, about a family of groups, namely: elementary abelian group of prime-fourth order.
View group cohomology of group families | View other specific information about elementary abelian group of prime-fourth order

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

$E_{p^4} = (\mathbb{Z}/p\mathbb{Z})^4 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$

## Particular cases

Value of prime $p$ elementary abelian group of prime-cube order cohomology information
2 elementary abelian group:E8 group cohomology of elementary abelian group:E8
3 elementary abelian group:E27 group cohomology of elementary abelian group:E27
5 elementary abelian group:E125 group cohomology of elementary abelian group:E125

## Homology groups for trivial group action

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

The homology groups over the integers are given as follows:

$\! H_q(E_{p^4};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{\frac{2q^3 + 15q^2 + 34q + 45}{24}}, & \qquad q = 1,3,5,\dots \\ (\mathbb{Z}/p\mathbb{Z})^{\frac{2q^3 + 15q^2 + 34q}{24}}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.$

The first few homology groups are given as follows:

$q$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$H_q$ $\mathbb{Z}$ $(\mathbb{Z}/p\mathbb{Z})^4$ $(\mathbb{Z}/p\mathbb{Z})^6$ $(\mathbb{Z}/p\mathbb{Z})^{14}$ $(\mathbb{Z}/p\mathbb{Z})^{21}$ $(\mathbb{Z}/p\mathbb{Z})^{35}$ $(\mathbb{Z}/p\mathbb{Z})^{49}$ $(\mathbb{Z}/p\mathbb{Z})^{71}$ $(\mathbb{Z}/p\mathbb{Z})^{94}$
rank of $H_q$ as an elementary abelian $p$-group -- 4 6 14 21 35 49 71 94