# Group cohomology of dicyclic groups

## Contents

This article gives specific information, namely, group cohomology, about a family of groups, namely: dicyclic group.
View group cohomology of group families | View other specific information about dicyclic group

We consider here the dicyclic group $\operatorname{Dic}_{4n}$ of degree $n$ and order $4n$.

## 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 for odd degree

The homology groups with coefficients in the ring of integers are as follows when the degree $n$ is odd.

$\! H_q(\operatorname{Dic}_{4n};\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad q = 0 \\ \mathbb{Z}/4\mathbb{Z}, & \qquad q \equiv 1 \pmod 4 \\ \mathbb{Z}/4n\mathbb{Z}, & \qquad q \equiv 3 \pmod 4 \\ 0, & \qquad q \ne 0, q \ \operatorname{even}\\\end{array}\right.$

### Over the integers for even degree

The homology groups with coefficients in the ring of integers are as follows when the degree $n$ is even.

$\! H_q(\operatorname{Dic}_{4n};\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad q = 0 \\ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, & \qquad q \equiv 1 \pmod 4 \\ \mathbb{Z}/4n\mathbb{Z}, & \qquad q \equiv 3 \pmod 4 \\ 0, & \qquad q \ne 0, q \ \operatorname{even}\\\end{array}\right.$