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 $S_{n}$ of degree $n$	degree $n=3$	group cohomology of symmetric groups
dihedral group $D_{2n}$ of degree $n$ and order $2n$	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:

$\!H_{p}(S_{3};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad p=0\\\mathbb {Z} /2\mathbb {Z} ,&\qquad p\equiv 1{\pmod {4}}\\\mathbb {Z} /6\mathbb {Z} ,&\qquad p\equiv 3{\pmod {4}}\\0,&\qquad p\neq 0,p\ \operatorname {even} \\\end{array}}\right.$

Over an abelian group

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

$H_{p}(S_{3};M)=\left\lbrace {\begin{array}{rl}M,&\qquad p=0\\M/2M,&\qquad p\equiv 1{\pmod {4}}\\\operatorname {Ann} _{M}(2)&\qquad p\equiv 2{\pmod {4}}\\M/6M,&\qquad p\equiv 3{\pmod {4}}\\\operatorname {Ann} _{M}(6),&\qquad p>0,p\equiv 0{\pmod {4}}\\\end{array}}\right.$

Here, $\operatorname {Ann} _{M}(2)$ denotes the 2-torsion subgroup of $M$ and $\operatorname {Ann} _{M}(6)$ 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:

$H^{p}(S_{3};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad p=0\\\mathbb {Z} /2\mathbb {Z} ,&\qquad p\equiv 2{\pmod {4}}\\\mathbb {Z} /6\mathbb {Z} ,&\qquad p\neq 0,p\equiv 0{\pmod {4}}\\0,&\qquad p\ \operatorname {odd} \\\end{array}}\right.$

Over an abelian group

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

$H^{p}(S_{3};M)=\left\lbrace {\begin{array}{rl}M,&\qquad p=0\\\operatorname {Ann} _{M}(2),&\qquad p\equiv 1{\pmod {4}}\\M/2M&\qquad p\equiv 2{\pmod {4}}\\\operatorname {Ann} _{M}(6),&\qquad p\equiv 3{\pmod {4}}\\M/6M,&\qquad p>0,p\equiv 0{\pmod {4}}\\\end{array}}\right.$

Here $\operatorname {Ann} _{M}(2)$ denotes the 2-torsion subgroup of $M$ and $\operatorname {Ann} _{M}(6)$ denotes the 6-torsion subgroup of $M$.