Group cohomology of symmetric group:S3

This article gives specific information, namely, group cohomology, about a particular group, namely: symmetric group:S3.
View group cohomology of particular groups | View other specific information about symmetric group:S3

Classifying space and corresponding chain complex

The homology and cohomology groups are the same as the respective homology and cohomology groups of the configuration space of three unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real vector space on the Topology Wiki.

Family contexts

Family name	Parameter value	General discussion of group cohomology of family
symmetric group	3	group cohomology of symmetric groups
dihedral group	order $2n=6$, degree $n=3$	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 ring of integers are as follows:

$H_p(S_3;\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& p=0\\ \mathbb{Z}/2\mathbb{Z},& p\equiv 1(\mathrm{mod}4)\\ \mathbb{Z}/6\mathbb{Z},& p\equiv 3(\mathrm{mod}4)\\ 0,& p\ne 0,p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\\ \end{array}$

Over an abelian group

The homology groups with coefficients in an abelian group are as follows:

$H_p(S_3;M)=\{\begin{array}{cc}M,& p=0\\ M/2M,& p\equiv 1(\mathrm{mod}4)\\ {\mathrm{A}}_{\mathrm{n}}(M)& p\equiv 2(\mathrm{mod}4)\\ M/6M,& p\equiv 3(\mathrm{mod}4)\\ {\mathrm{A}}_{\mathrm{n}}(M),& p>0,p\equiv 0(\mathrm{mod}4)\\ \end{array}$

Here, ${\mathrm{A}}_{\mathrm{n}}(M)$ denotes the 2-torsion subgroup of $M$ and ${\mathrm{A}}_{\mathrm{n}}(M)$ denotes the 6-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

The cohomology groups with coefficients in the ring of integers are as follows:

$H^p(S_3;\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& p=0\\ \mathbb{Z}/2\mathbb{Z},& p\equiv 2(\mathrm{mod}4)\\ \mathbb{Z}/6\mathbb{Z},& p\ne 0,p\equiv 0(\mathrm{mod}4)\\ 0,& p\mathrm{o}\mathrm{d}\mathrm{d}\\ \end{array}$

Over an abelian group

The cohomology groups with coefficients in an abelian group $M$ are as follows:

$H^p(S_3;M)=\{\begin{array}{cc}M,& p=0\\ {\mathrm{A}}_{\mathrm{n}}(M),& p\equiv 1(\mathrm{mod}4)\\ M/2M& p\equiv 2(\mathrm{mod}4)\\ {\mathrm{A}}_{\mathrm{n}}(M),& p\equiv 3(\mathrm{mod}4)\\ M/6M,& p>0,p\equiv 0(\mathrm{mod}4)\\ \end{array}$

Here ${\mathrm{A}}_{\mathrm{n}}(M)$ denotes the 2-torsion subgroup of $M$ and ${\mathrm{A}}_{\mathrm{n}}(M)$ denotes the 8-torsion subgroup of $M$.