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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

Contents

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

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.

Cohomology ring

Second cohomology group and extensions