# Group cohomology of 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:

$\! H_p(S_3;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/6\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even} \\\end{array}\right.$

### 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:

$H^p(S_3;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 2 \pmod 4 \\ \mathbb{Z}/6\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & \qquad p \ \operatorname{odd} \\\end{array}\right.$

### 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 $M$.