Group cohomology of dihedral groups

This article gives specific information, namely, group cohomology, about a family of groups, namely: group cohomology.
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 $n$).

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:

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

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:

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

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

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

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:

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

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