Group cohomology of dihedral group:D16

This article gives specific information, namely, group cohomology, about a particular group, namely: dihedral group:D16.
View group cohomology of particular groups | View other specific information about dihedral group:D16

This article describes the homology and cohomology group of dihedral group:D16, the dihedral group of order 16 and degree 8.

Family contexts

Family name Parameter value Information on group cohomology of family
dihedral group $D_{2n}$ of degree $n$, order $2n$ degree $n = 8$, order $2n = 16$ group cohomology of dihedral groups

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 with coefficients in the integers are given as follows:

$H_q(D_{16};\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & q = 0 \\ (\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & q \equiv 1 \pmod 4\\ (\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/8\mathbb{Z}, & q \equiv 3 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \mbox{ even }, q > 0 \\ \end{array}\right.$

The first few homology groups are as follows:

$q$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$H_q$ $\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/8\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $(\mathbb{Z}/2\mathbb{Z})^4$ $(\mathbb{Z}/2\mathbb{Z})^3$ $(\mathbb{Z}/2\mathbb{Z})^4 \oplus \mathbb{Z}/8\mathbb{Z}$ $(\mathbb{Z}/2\mathbb{Z})^4$