# Group cohomology of dihedral groups

View group cohomology of group families | View other specific information about group cohomology

We consider here the dihedral group $D_{2n}$ of order $2n$ and degree $n$ (i.e., its natural action is on a set of size $n$).

## Particular cases

Note that the case $n = 2$ is anomalous.

$n$ (degree) $2n$ (order) Group Cohomology information
2 4 Klein four-group group cohomology of Klein four-group
3 6 symmetric group:S3 group cohomology of symmetric group:S3
4 8 dihedral group:D8 group cohomology of dihedral group:D8
5 10 dihedral group:10 group cohomology of dihedral group:D10

## Homology groups for trivial group action

The below applies to $n \ge 3$. For $n = 2$, see group cohomology of Klein four-group.

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:

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

Note that a dihedral group of this sort is a finite group with periodic cohomology, which fits in with the alternative criterion for having periodic cohomology: every abelian subgroup is cyclic. In this case, the period on the homology groups with integer coefficients is 4.

### 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:

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

### Over an abelian group for odd degree

The homology groups with coefficients in an abelian group $M$ are as follows when the degree $n$ is odd:

$H_q(D_{2n};M) = \left \lbrace \begin{array}{rl} M, & \qquad q = 0 \\ M/2M, & \qquad q \equiv 1 \pmod 4 \\ \operatorname{Ann}_M(2) & \qquad q \equiv 2 \pmod 4 \\ M/2nM, & \qquad q \equiv 3 \pmod 4\\ \operatorname{Ann}_M(2n), & \qquad q > 0, q \equiv 0 \pmod 4 \\\end{array}\right.$

Here, $\operatorname{Ann}_M(2)$ denotes the 2-torsion subgroup of $M$ and $\operatorname{Ann}_M(2n)$ denotes the $2n$-torsion subgroup of $M$.

## Cohomology groups for trivial group action

FACTS TO CHECK AGAINST (cohomology group for trivial group action):
First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization
In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

### Over the integers for odd degree

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

$H^q(D_{2n};\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad q = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad q \equiv 2 \pmod 4 \\ \mathbb{Z}/2n\mathbb{Z}, & \qquad q \ne 0, q \equiv 0 \pmod 4\\ 0, & \qquad q \ \operatorname{odd} \\\end{array}\right.$

### Over an abelian group for odd degree

The cohomology groups with coefficients in an abelian group $M$ are as follows when the degree $n$ is odd:

$H^q(D_{2n};M) = \left \lbrace \begin{array}{rl} M, & \qquad q = 0 \\ \operatorname{Ann}_M(2), & \qquad q \equiv 1 \pmod 4 \\ M/2M & \qquad q \equiv 2 \pmod 4 \\ \operatorname{Ann}_M(2n), & \qquad q \equiv 3 \pmod 4\\ M/2nM, & \qquad q > 0, q \equiv 0 \pmod 4 \\\end{array}\right.$

Here $\operatorname{Ann}_M(2)$ denotes the 2-torsion subgroup of $M$ and $\operatorname{Ann}_M(2n)$ denotes the $2n$-torsion subgroup of $M$.