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:

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

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:

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

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 $M$.